text:
\section{}

Deep learning surrogate models are being increasingly used in accelerating scientific simulations as a replacement for costly conventional numerical techniques. However, their use remains a significant challenge when dealing with real-world complex examples. In this work, we demonstrate three types of neural network architectures for efficient learning of highly non-linear deformations of solid bodies. The first two architectures are based on the recently proposed CNN U-NET and MAgNET (graph U-NET) frameworks which have shown promising performance for learning on mesh-based data. The third architecture is Perceiver IO, a very recent architecture that belongs to the family of attention-based neural networks--a class that has revolutionised diverse engineering fields and is still unexplored in computational mechanics. We study and compare the performance of all three networks on two benchmark examples, and show their capabilities to accurately predict the non-linear mechanical responses of soft bodies.

\tiny
 \fontsize{8}{11}\helveticabold { \section{Keywords:} Surrogate Modeling, Deep Learning, CNN U-NET, Graph U-Net, Perceiver IO, Finite Element Method}
\end{abstract}

\section{Introduction}

Fast or real-time predictions of mechanical responses are fundamental for multiple engineering applications. Notable examples can be found in the field of robotics \citep{rus2015design, choi2021use} or medical simulations \citep{courtecuisse2014real,cotin,bui,MAZIER2021110645}, that are promising progress in personalized medicine and advanced computer-assisted- or robotic surgery~\citep{chen2020deep,Dennler2021}. In many cases, the necessary accuracy of prediction requires complex models that can be generally expressed through partial differential equations and solved numerically, e.g., with the finite element method (FEM). However, those high-fidelity computational models are usually too slow for real-time purposes, and the necessary speed-up is achieved by developing surrogate models.

At the same time, the XXI century has given birth to an immense amount of measurement data, which is also available as public datasets in various scientific domains, including structural mechanics and material science \citep{zakutayev2018open,ELOUNEG2022107835,GHOLAMALIZADEH2022107009}. The abundant data availability and rapid growth in computational resources have given rise to machine learning (ML) techniques \citep{butler2018machine,bock2019review,schleder2019dft} which are gaining increasing importance in solving forward and inverse engineering problems. This includes data-driven approaches aiming to enable modeling \citep{ma12162574}, to accelerating computationally costly direct numerical simulations \citep{wirtz2015surrogate, CAPUANO2019363,weerasuriya2021gaussian} or even to discovering new material laws \citep{LIU2017159,euclid}.

Within the class of machine learning methods to surrogate- and data-driven modeling, deep learning (DL) approaches have tasted great success because of their ability to efficiently extract complex relationships present in the underlying data. DL models have been successfully employed for a range of tasks in diverse fields such as material science, computational mechanics, computer vision, etc., \citep{jha2018elemnet,voulodimos2018deep,OISHI2017327,choudhary2022recent}. In particular, also solid mechanics has witnessed an increasing use of DL surrogate models as a replacement to costly direct numerical simulations \citep{mianroodi2021teaching,ABUEIDDA2021102852}. Deep learning techniques rely on deep artificial neural networks (deep ANNs, or DNNs). DNNs need to be trained on a sufficiently large amount of data, which is a computationally costly process. However, once trained, the predictions of DL models are computationally extremely efficient.

It is often difficult to obtain the necessary amount of training data when it  originates from physical experiments. This can be due to multiple factors, such as high costs, risks \& difficulties associated with the experiments, or data privacy clauses.
There are two possible approaches to deal with the scarcity of experimental data. The first approach relies on enhancing the DL model with the information on underlying physics -- an approach popularly termed as Physics Informed Neural Networks (PINN) \citep{earlyPINN,MAO2020112789,SAMANIEGO2020112790,alban}. The second approach includes the underlying physics implicitly, through high-fidelity simulations done \emph{in silico} to provide the necessary amount of synthetically generated data, which has shown to be useful in various applications \citep{le2017using,kim2022how,Pfeiffer2019,aydin2019general,soumi}. In this work, we will follow the latter approach and will focus on DL surrogate models that are trained on synthetically generated data from finite element simulations in non-linear elasticity.

One of the most important aspects that will be studied in this work is the architecture of deep neural networks. The majority of DL approaches that are present in the literature are based on fully connected networks, which are inefficient and prone to overfitting when applied to high-dimensional inputs. If such large inputs are structured, they fall under the umbrella of the geometric deep learning (GDL) paradigm \citep{bronstein2021geometric}--a concept that has been gaining increasing interest in recent years. In this work, we will compare three architectures that can efficiently deal with high-dimensional structured inputs: convolutional neural networks (CNNs), graph neural networks (GNNs), and attention-based networks.

Convolutional neural networks (CNNs) are known to outperform traditional fully-connected ANNs, and this has been demonstrated in various domains, including physics-based simulations \citep{10.1145/2939672.2939738,vasilis,doi:10.1063/5.0077723,DESHPANDE2022115307}. CNNs work on the principle of parameter sharing and local convolution operations, which enables efficient training on large inputs. Their disadvantage is that the inputs/outputs of CNNs are restricted to grid inputs, such as images, videos, or structured FE meshes. However, CNNs have found their successors, the graph neural networks (GNNs), that can work with any structure of inputs/outputs.

Graph-based approaches leverage the topological information of the input to perform local operations in the respective neighborhood only and can learn efficiently on unstructured data. Recently GDL methods have shown promising performance for their applications in the mechanics field as well \citep{battaglia2018relational,VLASSIS2020113299,pfaff2021learning,vasilisgraph,sebstornich}. More recently \citep{magnet} proposed MAgNET, a novel graph U-Net framework for efficiently learning on mesh-based data. In this work we utilise it to accurately predict non-linear deformations of solids.

\sd{

}

Attention-based approaches, similar to human cognitive attention, work by allowing the DL model to focus on certain parts of the input data that are relevant to the task at hand. This is done through a fully trainable process that, without the need to introduce topological information or enforce structural restrictions, allows the neural network to extract dependencies from throughout the whole input domain. This type of approach has led to significant strides in a wide range of areas, starting from computer vision \citep{xu2015show} to natural language processing \citep{devlin2018bert,baevski2020wav2vec}, as well as becoming the basic building block of the Transformer architecture \citep{vaswani2017attention}. Recently the Perceiver~IO~\citep{jaegle2022perceiver}, a new type of architecture that builds upon Transformers, has been proposed as a general-purpose model that can handle data from arbitrary settings. Since Perceiver IO has been shown to achieve several state-of-the-art results without the need for problem-specific architecture engineering, we will compare its performance on non-linear deformation prediction of solids based on mesh data against the previously discussed models.

To summarise, in this work we will compare three architectures: two architectures presented in our earlier works, i.e., CNN U-Net framework~\citep{DESHPANDE2022115307}, and MAgNET framework~\citep{magnet}, as well as the attention-based architecture, Perceiver~IO \citep{jaegle2022perceiver}, which hasn't been explored for its applications in mechanics yet. We show the capabilities of three frameworks by learning on non-linear FEM datasets and by cross-comparing their performance. In Section~\ref{sec: methodlogy}, we will introduce the three DNN architectures, in Section~\ref{sec: results}, we will provide the study of their performance, and in Section~\ref{sec: conclusion} we will summarize the results and discuss future directions.

\section{Method}\label{sec: methodlogy}

\begin{figure}[t!]
\begin{center}
\includegraphics[width=1\textwidth]{PDF_examples/outline.pdf
\end{center}
\caption{Outline of the neural network surrogate frameworks for predicting body deformations. (Left) Training datasets for structured and arbitrary mesh cases are generated by using a non-linear FEM solver. (Middle) Proposed neural network frameworks are trained on these datasets. For structured mesh case, all NN frameworks are used while for arbitrary unstructured meshes only MAgNET and Perceiver~IO networks are used. (Right) Trained networks are then used as surrogate models to predict the deformation of bodies under unseen forces.}\label{fig: outline}
\end{figure}

As mentioned in the introduction, in this paper we propose different types of deep neural network (DNN) frameworks capable of being used as surrogate model replacements for computationally costly non-linear FEM solver. The proposed DNN frameworks are trained on force-displacement FEM datasets. Once trained, DNN surrgoate models are fast and accurate in simulating mechanical responses of bodies subjected to external forces. The overall outline of the paper is illustrated in the Figure~\ref{fig: outline}

Any input mesh can be categorised into either a structured mesh or an arbitrary unstructured mesh. In this work we introduce three types of DNN frameworks. First is the CNN U-Net network which can only be used in an straight forward manner for structured meshes. Where as MAgNET and Perceiver~IO networks are capable of handling any arbitrary mesh inputs. All these frameworks are discussed in detail in following subsections.

\subsection{DNN frameworks for predicting mechanical deformations}

Below we introduce three different types of neural network architectures which can efficiently predict non-linear deformations of bodies subjected to external traction and body forces. All the proposed DNN frameworks directly operate on the finite element mesh data thereby making it very convenient to be used as surrogate models in place of conventional FEM solver. The first two i.e. CNN U-Net and MAgNET belong to the family of U-Net architecture while Perceiver IO is based on Transformer-type attention, see Table~\ref{tab:frameworks}.

\bgroup
\def1.5{1.4}
\begin{table}[h!]
    \centering
    \begin{tabular}{c|c|c}
        Framework & Type & Mesh support \\
        \hline
        CNN & U-Net (convolution operation) & structured    \\
        MAgNET  & Graph U-NET (MAg operation)  &arbitrary   \\
        Perceiver~IO & Transformer (attention mechanism) & arbitrary
    \end{tabular}
    \captionsetup{justification=centering}
    \caption{Properties of deep neural network architectures studied in this work.}
    \label{tab:frameworks}
\end{table}
\egroup

\subsubsection{CNN U-Net}
CNNs were originally proposed for performing classification and regression tasks on image, video like data but lately are even being used for generic inputs such as mesh data which is crucial to many scientific applications. In particular, U-Net like architectures have shown great potential in learning on large-scale inputs and lately have been successfully used for simulating mechanical responses of materials as well \citep{mianroodi2021teaching,DESHPANDE2022115307}. The name U-Net comes from the particular U-shaped architecture which involves a series of convolutional and pooling operations. Convolutional layers are responsible for non-linear transformations whereas pooling enables learning through low-fidelity representation thus making the network capable of learning on high-dimensional inputs. Experiments presented in this work are carried out by using the CNN U-Net framework proposed by \citep{DESHPANDE2022115307}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\textwidth]{PDF_examples/CNN_architecture.pdf
\end{center}
\caption{ Schematic of CNN architecture used for generic structured 2D mesh inputs.}\label{fig: CNN schematic}
\end{figure}

One major limitation of the CNN is that it cannot straightforwardly accommodate unstructured mesh inputs. To overcome this issue, the simplest approach embeds a structured grid on unstructured meshes with a naive mapping between unstructured and grid node values \citep{U-Mesh}. While more sophisticated approaches are proposed to make unstructured meshes compatible to be used with CNN framework \citep{immerserdboundary}. However, they perform poorly on complicated geometries and come with an associated preprocessing cost; they are not considered in the scope of this work.

\subsubsection{MAgNET}

In an attempt to generalise CNN to arbitrary unstructured meshes, very recently \citep{magnet} proposed the MAgNET framework. MAgNET architecture belongs to the family of graph U-Net architectures and it is proposed for efficient learning on mesh structured data. MAgNET directly accepts arbitrary mesh inputs (such as forces/stresses/displacements of nodes in the mesh) values thus making it very convenient to be used with existing numerical solvers.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\textwidth]{PDF_examples/magnet_schematic.pdf
\end{center}
\caption{Schematic of the MAgNET architecture used in this work. It takes external forces on arbitrary mesh as an input to gives mesh displacements as output.   }\label{fig: magnet_schematic}
\end{figure}

MAgNET relies on the so-called MAg layer (Multichannel Aggregation layer) which is capable of learning nonlinear transformations between input and output data existing in the mesh format. MAg extends the concept of local operations in convolution layers to arbitrary mesh inputs by performing aggregation of nodal feature values in the respective neighborhood nodes only. It leverages the topology of inputs and performs learnable local aggregations with heterogeneous window sizes as opposed to the fixed-size window in the case of CNN. While its graph pooling/unpooling layers enable efficient learning on large-dimension inputs through reduced graph representation.

\subsubsection{Perceiver IO}

The Perceiver IO architecture, \citep{jaegle2022perceiver}, was developed with the goal of achieving a DL scheme that can easily integrate and transform arbitrary information for arbitrary tasks. This architecture employs an attention encoder that maps inputs from a wide range of modalities to a fixed-size latent space using cross-attention, this latent space is then further processed using self-attention as an usual Transformer and decoded into the output domain via cross-attention. This process allows the network to scale to large and multi-modal data since it decouples the bulk of the network's processing from the size and modality-specific details of the input. In this work, we will leverage this property and use Perceiver IO to learn non-linear deformations on unstructured meshes without adding any information or restrictions about how to treat the underlying data structure. During training, Perceiver IO automatically learns the important dependencies that exist in the input domain, composed of arbitrarily unstructured mesh data, and transforms them into the corresponding output which consists of displacement data.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\textwidth]{PDF_examples/Perceiver_schematic_modified.png
\end{center}
\caption{Schematic of the Perceiver IO architecture, \citep{jaegle2022perceiver}, used for external forces on arbitrary mesh as inputs and mesh displacement as outputs.}\label{fig: Perceiver schematic}
\end{figure}

\subsection{Input/output \& training of DNN surrogate models} \label{sec: inouttraining}

As motivated in Section~\ref{sec: methodlogy}, the proposed DNN frameworks are trained on the force-displacement datasets. Let us denote the neural network in consideration as $h$, it is parameterised by trainable parameters, $\bfs{\theta}$. In all the cases, $h$ accepts external forces, \bfm{f}, on all degrees of freedom (dofs) of mesh as the input. And as an output it predicts displacement vector, \bfm{u}, of all dofs (same size as the input). I.e. $h: \bfm{f}\rightarrow\bfm{u} $.

In the case of CNN and MagNET, forces associated with X, Y, Z degrees of freedom are kept in different channels. For instance, in the case of CNN U-Net, X, Y directional forces on a 2D quad mesh with  $n_x \times n_y$ nodes are fed to the network as a $2 \times n_x\times n_y$ tensor. For MAgNET, they are provided as a single dimensional tensor of shape $2 \cdot n_x \cdot n_y$, with X and Y directional forces concatenated together (each representing a different channel). On the other hand, Perceiver IO inputs are first kept as a single array of size $2 \cdot n_x \cdot n_y$ to which we apply 1 × 1 convolution kernels to project it to a tensor of shape $2 \cdot n_x \cdot n_y \times 256$, adding $256$ channels. Finally, to this channel dimension we concatenate trainable 1D positional embeddings, thus leaving us with a tensor of shape $2 \cdot n_x \cdot n_y \times 512$ that we will use as an input for the network.

Now for a given training dataset $\{(\bfm{f}_1,\bfm{u}_1),...,(\bfm{f}_N,\bfm{u}_N)\}$, $h$ is trained by minimizing mean squared error between true and predicted values as to get optimised parameters $\bm{\theta}^{*}$ as:

\begin{equation}
      \bm{\theta}^{*} = \text{arg}\underset{\bm{\theta}}{\text{min}}~\frac{1}{N}\sum_{i=1}^{N} \norm{h(\bfm{f}_i,\bm{\theta})-\bfm{u}_i}_2^{2}
     \label{eq:lossDeterm}
\end{equation}

Performance of $h$ (i.e. the neural network in consideration) over respective test dataset $\{(\bfm{f}_1,\bfm{u}_1),...,(\bfm{f}_M,\bfm{u}_M)\}$ is measured in terms of mean absolute error which is computed for each example ($e_m$) and for the entire test set ($\Bar{e}$) as follows:

\begin{equation}
     e_{m} = \frac{1}{\mathcal{F}}\sum_{i=1}^{\mathcal{F}}{|h(\bfm{f}_m)^{i} - \bfm{u}_m^{i}|}, \qquad
     \Bar{e} =\frac{1}{M} \sum_{m=1}^{M} e_{m}, \qquad
     \sigma(e) = \sqrt{\frac{1}{M-1}\sum_{m=1}^{M} \left(e_{m} - \Bar{e} \right)^2}
     \label{eq: test loss}
 \end{equation}

Where $\mathcal{F}$ stands for the number of dofs of the mesh.

\section{Results}
\label{sec: results}

We validate proposed neural network frameworks on two examples, representing a 2D and a 3D problem respectively. For the 2D example, a structured mesh is considered so that all frameworks including CNN can be applied to it. Whereas for the 3D example an arbitrary unstructured mesh is considered.

\subsection{Generation of hyperelastic FEM training data}\label{sec: data generation}

As motivated in the methodology section, training datasets of non-linear displacement solutions are generated by applying random traction and body forces on the given discretisation. And the pairs of input force and output displacement solutions are then fed to train different types of neural networks. The proposed neural network frameworks are validated on two examples, both following Neo-Hookean hyperelastic law. To avoid the divergence of the non-linear FEM solver, both traction and body forces are applied in incremental load steps. All the computations are performed using AceFEM framework \citep{acegen}.

For the 2D case, a rectangular domain made of soft material and discretized by $8\times32$ mesh with 217 quad elements is considered. It is constrained at 4 corner nodes as shown in Figure~\ref{fig:2dschematic}. It is subjected to traction forces of random magnitude, location, and direction in the region prescribed by the pink line. Body forces are ignored in this case. Table~\ref{tab:datasets} provides detailed information about datasets including the material properties and external force ranges used for the generation of 2D and 3D datasets. For the 2D case, a lower range of Y-direction force density is chosen since it doesn't contribute much to generating large deformation solutions.

In the 3D case, a continuum toy elephant model is discretised with 6627 tetrahedron elements. It is subjected to fixed boundary conditions by constraining nodes on the bottom region of the legs. Body forces in random magnitude and direction are applied in the transverse directions (see Figure~\ref{fig:3dscehmatic}) to generate datasets of force-displacement pairs. External tractions are ignored in this case.

\begin{table}[h!]
    \centering
    \begin{tabular}{l|R|R|L|R}
        Case& Material properties ~~
        ($E$ Pa, $\nu$) & N.of FEM DOFs ($\mathcal{F})$ & External traction/body force density range & Dataset size (train + test)\\
        \hline
        2D domain& 100, 0.3 & 512 & $f_x$ = -24 to 24 $\text{N}/\text{m}$, $f_y$ = -8 to 8 $\text{N}/\text{m}$  & $7124 + 372$\\
        3D elephant& 3 E6, 0.4 & 5835 & $b_x, b_z$ = -0.35 to 0.35 $\text{N}/\text{kg}$, $b_y$=0 $\text{N}/\text{kg}$ & $7600 + 400$
    \end{tabular}
    \captionsetup{justification=centering}
    \caption{Desciption of FEM datasets}
    \label{tab:datasets}
\end{table}

\setcounter{figure}{5}
\setcounter{subfigure}{0}
\begin{subfigure}
\centering
\setcounter{figure}{5}
\setcounter{subfigure}{0}
    \begin{minipage}[b]{0.14\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/2D_schematic.pdf}
        \caption{}
        \label{fig:2dschematic}
    \end{minipage} \hspace{0.15\textwidth}
\setcounter{figure}{5}
\setcounter{subfigure}{1}
    \begin{minipage}[b]{0.53\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/elephant_scehmatic_final.pdf}
        \caption{}
        \label{fig:3dscehmatic}
    \end{minipage}
\setcounter{figure}{5}
\setcounter{subfigure}{-1}
    \caption{Schematics of dataset generation for \textbf{(a)} 2D example subjected to external traction forces. \textbf{(b)} 3D example subjected to external body forces. External tractions and body forces are indicated with pink arrows.}
    \label{fig: schematic}
\end{subfigure}

\subsection{Implementation details}

As introduced in the methodology section, CNN and MAgNET frameworks belong to the family of U-Net architectures, while Perceiver IO leverages Transformer-style attention. It has to be noted that all the frameworks are robust, and do not need fine hyperparameter tuning.

\textbf{2D case}: For the CNN U-Net, a 4-level architecture with 3 max-pooling/upsampling operations is used. At each level, two convolution layers with $3\times3$ filters are applied with 64, 128, 256, and 256 channels at respective levels. In the case of MAgNET, a 5-level graph U-Net architecture with 4 graph pooling/unpooling operations is used. At each level 2 MAg layers (with $A^2$ adjacency, refer \citep{magnet}) are applied with 8, 16, 16, 32, and 32 channels at respective levels.  For both 2D and 3D case MAgNET architectures, the seed of the first graph pooling operation is chosen by grid search, while seeds for other pooling layers are kept constant. This is done to have the maximum possible coarsened representation of the lowest-level graph. This ensures propagating boundary condition information with a minimum number of MAg operations at the lowest level. Both CNN and MAgNET are trained for 5000 epochs with a batch size of 4.

In the case of Perceiver IO we defined 512 inputs (standing for dofs of the example) with a total embedding size of 512 following the procedure detailed in Section~\ref{sec: inouttraining}. We also selected a total of 128 latent arrays of dimension 210 for performing cross-attention in the encoder, self attention in latent space and inputs for the decoder. For the decoder's output query array we used an index dimension of 512, which defines the size of the outputs, and a channel dimension of 210 equal to the dimension size of the latents. We used a total of 3 blocks for the latent array processing, with 2 self-attention layers per block and 2 self-attention heads per layer. Both the encoder and decoder worked with 2 cross-attention heads each. The selection of these hyper-parameters was determined in a coarse exploratory fashion, with the goal of reducing the number of network parameters while maintaining its performance. Perceiver IO is trained with a batch size of 16 for 32000 epochs.

\textbf{3D case}: For the MAgNET, 7-level graph U-Net architecture is used with 6 graph pooling/unpooling operations. Again at each level, 2 MAg layers (with $A^2$ adjacency) are applied with 6, 6, 6, 12, 12, 24, 24 channels at respective levels. The complex topology of this particular mesh demands more number graph pooling layers, this ensures propagating boundary condition information with a minimum number of MAg operations at the lowest level. On the other hand, the only change with respect to the 2D case for Perceiver IO is an increase of the input and output dimension from 512 to 5835 in both the encoder and decoder. MAgNET architecture for this case is trained for 600 epochs with a batch size of 4, whereas Perceiver IO is trained for 1500 epochs with a batch size of 16.

CNN U-Net and MAgNET networks are trained using the Adam optimizer \citep{adam}, whereas Perceiver IO is trained using AdamW optimizer \citep{adamw} as implemented in the original paper. CNN and MAgNET are implemented using TensorFlow \citep{tensorflow2015-whitepaper}, while Perceiver IO is implemented using PyTorch \citep{pytorch}. All the implementations in this work are performed using HPC facilities of the University of Luxembourg \citep{ULHPC}.

\subsection{Performance on unseen examples}

Proposed DNN frameworks are trained and tested on the datasets generated as illustrated in Section~\ref{sec: data generation}. The maximum nodal displacement for the 2D case is 0.35 m and for the 3D case it is 140.04 m, i.e. we compute displacements of all the nodes for every single example, and then choose particular examples for which the maximum nodal displacement is observed. Table~\ref{tab: test metrics} summarises the performance of neural networks on the two test datasets.  It shows that all three networks are capable of predicting mechanical deformation responses with a very low error.

In particular, we observed that Perceiver IO gave lower error metrics for the 2D case, which has relatively low dimensional input. But as the size and complexity of the mesh increased in the 3D case, MAgNET performed better. A possible explanation for this behaviour is mentioned in the next subsection.

\bgroup
\def1.5{1.5}
\begin{table}[h]
\begin{center}
 \begin{tabular}{l| c | c | c | c| c}
 Example & Framework & $M_{\text{te}}$ & $\Bar{e}$ [m] & $\sigma(e)~[\text{m}]$& $e_{\text{max}}~[\text{m}]$  \\ [0.7ex]
 \hline
\multirow{3}{*}{2D} & CNN & \multirow{3}{*}{372} & 0.16 E-3 & 0.4 E-4 & 0.0018  \\
& MAgNET &  & 0.28 E-3 & 1.0 E-4 &  0.0214  \\
& Perceiver IO &  & 0.03 E-3 & 0.1 E-4  &  0.0013  \\
 \hline
 \multirow{2}{*}{3D}& MAgNET & \multirow{2}{*}{400}  & 1.05 E-2 & 2.0 E-3 & 0.271   \\
 & Perceiver IO &  & 1.50 E-2 & 5.0 E-3  & 0.590  \\

\end{tabular}
\end{center}
\caption{Error metrics over the test set using the proposed NN frameworks. $M$ stands for the number of test examples, and $\Bar{e}$, $\sigma(e),e_{\text{max}}$ are error metrics defined in Section~\ref{sec: inouttraining}.}
\label{tab: test metrics}
\end{table}

\subsection{Training and inference of DNN frameworks}

First, we compare the training convergence of proposed DNN frameworks by comparing the mean square loss plots for both 2D and 3D cases. Figure~\ref{fig: 2dloss} shows that for the relatively smaller dimension inputs as in the 2D case, Perceiver IO is observed to learn more efficiently when compared to CNN and MAgNET. Whereas  MAgNET is more efficient in the case of complex meshes and higher dimensional inputs such as in the 3D case. As the size and complexity of mesh further increases, Perceiver IO is not robust as MAgNET for efficient learning on such data. We observed that Perceiver IO failed to learn on the data with input dimension higher than $10^4$.

A possible interpretation of this behavior is that in the case of MAgNET topological information is externally provided through the adjacency matrix. Hence MAgNET can efficiently learn by leveraging inter-dependencies between different nodal feature values in the data. On the other hand Perceiver IO implicitly learns the nodal data dependencies and as the size of the input data increases, the task to find these inter-dependencies gets more difficult. Evidence of this behavior can be seen in Figure~\ref{fig: convergence}, where it is clear that Perceiver IO is optimizing through a much more complex objective function with a higher density of local minimas.

\setcounter{figure}{6}
\setcounter{subfigure}{0}
\begin{subfigure}
\centering
\setcounter{figure}{6}
\setcounter{subfigure}{0}
    \begin{minipage}[b]{0.45\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/2D_plot.png}
        \caption{}
        \label{fig: 2dloss}
    \end{minipage} \hspace{0.05\textwidth}
\setcounter{figure}{6}
\setcounter{subfigure}{1}
    \begin{minipage}[b]{0.45\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/Elephant_loss_plot.png}
        \caption{}
        \label{fig:3dloss}
    \end{minipage}
\setcounter{figure}{6}
\setcounter{subfigure}{-1}
    \caption{Training convergence for the proposed neural network frameworks for the \textbf{(a)} 2D case \textbf{(b)} 3D case.}
    \label{fig: convergence}
\end{subfigure}

Once trained, proposed DNN frameworks are fast at the inference stage while predicting unseen examples. Table~\ref{tab: training and test times} provides training and inference time (for a single test example) for all three frameworks, for comparison FEM solution time is also provided. In particular, Perceiver IO takes a much longer time during the training phase but is extremely fast at the inference stage. It could make predictions on both small scale (2D) and large scale (3D) inputs in almost similar time. It has to be noted that to ensure the convergence of the iterative solver, the non-linear FEM problem is solved with incremental load steps. Hence the solution time for FEM increases with the magnitude of external force. Whereas trained DNN frameworks take almost similar time at the inference stage irrespective of external force magnitudes.

\bgroup
\def1.5{1.5}
\begin{table}[h]
\begin{center}
 \begin{tabular}{l| R | R | R | R | R }
 Example & Framework & N. parameters ($\times$ E6)& Training time (hrs) & Inference time (s) & FEM solver time (s) \\ [0.7ex]
 \hline
\multirow{3}{*}{2D} & CNN U-Net & 4.8 & 10 & 0.021 & \multirow{3}{*}{0.6}\\
& MAgNET & 4.5 & 77 & 0.050  &\\
& Perceiver IO & 1.9 & 240 & 0.006 &  \\
 \hline
 \multirow{2}{*}{3D}& MAgNET& 33.9 & 79 & 0.217& \multirow{2}{*}{2.5} \\
 & Perceiver IO & 4.4 &  144 &  0.006 &\\

\end{tabular}
\end{center}
\caption{Comparison of training and inference times for all the three networks implemented in this work.}
\label{tab: training and test times}
\end{table}

\subsection{Qualitative analysis of individual examples}

In this section, we analyze deformations of individual examples by giving a qualitative comparison of predictions obtained using different networks. In particular, we analyze test examples with the maximum nodal displacement for 2D as well as the 3D case. In both cases, we plot nodal error contours standing for the absolute difference between the DNN prediction and the true FEM solution.

\subsubsection{2D case}

The analyzed example in the Figure~\ref{fig: 2d viz} stands for the maximum nodal displacement example in the 2D test dataset. The node indicated by the green dot has the maximum nodal displacement of 0.35 m. While the pink arrows represent corresponding nodal forces for the line density force applied on those four nodes.

Figure~\ref{fig: 2d viz} shows absolute error counters of nodal displacements predicted using proposed DNN frameworks when compared with the FEM solution.  All the proposed neural network frameworks can accurately predict the deformed mesh. Percentage prediction error (when compared to the true FEM solution) for the green node is  0.12\%, 0.46\%, 0.09\% using CNN, MAgNET, and Perceiver IO network respectively.  We observed that for the small-scale inputs Perceiver IO is observed to make better predictions when compared to the other two networks. This advantage likely comes from the network leveraging its capability of learning long-range correlations more accurately. This stems from the fact that Perceiver's inputs are not constrained by any topological assumption.

\setcounter{figure}{0}
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\begin{subfigure}
\centering
\setcounter{figure}{7}
\setcounter{subfigure}{0}
    \begin{minipage}[b]{0.26\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/2dclothcnn_11_11_t50_error.pdf}
        \caption{CNN}
        \label{fig:2dcnnerrror}
    \end{minipage} \hspace{0.08\textwidth}
\setcounter{figure}{7}
\setcounter{subfigure}{1}
    \begin{minipage}[b]{0.26\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/2dclothmag_11_11_t50_error.pdf}
        \caption{MAgNET}
        \label{fig:2dmagneterror}
    \end{minipage} \hspace{0.08\textwidth}
\setcounter{figure}{7}
\setcounter{subfigure}{2}
    \begin{minipage}[b]{0.26\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/2D_perceiver_t50_error.pdf}
        \caption{Perceiver IO}
        \label{fig: 2dperciever}
    \end{minipage} \hspace{0.08\textwidth}
\setcounter{figure}{7}
\setcounter{subfigure}{-1}
    \caption{Prediction error for different neural network frameworks when compared to true FEM solution, plotted on the deformed mesh (obtained using the same framework). Force with line density of (-21.6645,-2.99384) N is applied as shown with pink arrows. True displacement of the green node is 0.35m. Solution error \textbf{(a)} using CNN \textbf{(b)} using MAgNET (c) using Perceiver IO.}
    \label{fig: 2d viz}
\end{subfigure}

\subsubsection{3D case}

The 3D case considered in this paper depicts a real-world complex geometry. Tetrahedron discretisation of the continuous 3D domain is irregular and even the mesh topology is complex. The fixed nodes are topologically far from the tip of the elephant trunk, thus making it challenging to communicate the boundary condition information. We show that both MAgNET and Perceiver IO can learn on such complex real-world examples efficiently.

Again we consider the test example with maximum nodal displacement. Figure~\ref{fig: 3d viz} shows that both MAgNET and Perceiver IO solutions are able to predict non-linear mesh deformations accurately. We further analyse the absolute nodal error counters for both predictions by plotting them on the deformed meshes predicted using respective DNN frameworks. Both front (Figure~\ref{fig:3d magnet error(front)}-\ref{fig:3d perceiver error(front)}) and side views (Figure~\ref{fig:3d magnet error(side)}-\ref{fig:3d perceiver error(side)}) indicate low prediction errors for both frameworks. The green node (at the tip of the ear) shown in Figure~\ref{fig:3d magnet error(front)}-\ref{fig:3d perceiver error(front)} has a maximum nodal displacement of 140.04 m for this example. The percentage prediction error (when compared to the true FEM solution) for this green node is 0.01\%, and 0.09\% using MAgNET and Perceiver IO networks respectively. For this demanding case, MAgNET is observed to make better predictions compared to Perceiver IO. However owing to the lesser number of trainable parameters, Perceiver IO is much faster at the inference stage.

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     \hspace{-0.09\textwidth}
    \begin{minipage}[b]{0.36\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/3dnormal_11_11_t269_mesh.pdf}
        \caption{MAgNET prediction}
        \label{fig:3d magnet mesh}
    \end{minipage} \hspace{0.15\textwidth}
\setcounter{figure}{8}
\setcounter{subfigure}{1}
    \begin{minipage}[b]{0.36\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/3dperceiver_t269_mesh.pdf}
        \caption{Perceiver IO prediction}
        \label{fig:3d perceiver mesh}
    \end{minipage} \hspace{0.02\textwidth}
\setcounter{figure}{8}
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    \begin{minipage}[b]{0.46\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/3dnormal_11_11_t269_errorside.pdf}
         \caption{MAgNET error (side view)}
        \label{fig:3d magnet error(side)}
    \end{minipage} \hspace{0.04\textwidth}
\setcounter{figure}{8}
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    \begin{minipage}[b]{0.46\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/3dperceiver_t269_errorside.pdf}
         \caption{Perceiver IO error (side view)}
        \label{fig:3d perceiver error(side)}
    \end{minipage} \hspace{0.08\textwidth}
\setcounter{figure}{8}
\setcounter{subfigure}{4}
    \begin{minipage}[b]{0.46\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/3dnormal_11_11_t269_errorfront2.pdf}
        \caption{MAgNET error (front view)}
        \label{fig:3d magnet error(front)}
    \end{minipage} \hspace{0.04\textwidth}
\setcounter{figure}{8}
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    \begin{minipage}[b]{0.46\textwidth}
        \includegraphics[width=\linewidth]{PDF_examples/3dperceiver_t269_errorfront2.pdf}
        \caption{Perceiver IO error (front view)}
        \label{fig:3d perceiver error(front)}
    \end{minipage} \hspace{0.08\textwidth}
\setcounter{figure}{8}

\setcounter{subfigure}{-1}
    \caption{Deformation of elephant subjected to external body force density (0.34, 0.0, 0.35) $\text{N}/\text{kg}$. \textbf{(a)\&(b)} Deformed meshes using MAgNET~(dark blue) and Perceiver IO~(sky blue) respectively, for comparison FEM mesh is presented in red. The rest position is indicated with gray mesh. \textbf{(c)\&(d)} Nodal errors when compared to the FEM solutions, plotted on the deformed meshes for MAgNET and Perceiver IO solution respectively. \textbf{(e)\&(f)} Front view of mesh error contours for MAgNET and Perceiver IO respectively. The true displacement of the green node is 140.04 m.}
    \label{fig: 3d viz}
\end{subfigure}

\section{Conclusion}
\label{sec: conclusion}

In this work, we demonstrated the capabilities of three promising deep neural network (DNN) frameworks for accurate and fast predictions of non-linear deformations of solid bodies. We compared their performance on two benchmark examples, in which data was generated by the finite element method. Although we only tested the frameworks for the Noe-Hoohean material model, they are compatible with more general hyperelastic models, such as  Mooney–Rivlin or Ogden models. As such, they promise to be used as surrogate models for non-linear computational models in mechanics.

The comparison included two very recent DNN frameworks, MAgNET and Perceiver~IO, that are naturally able to work with arbitrarily structured data at inputs/outputs, including complex finite element meshes that originate from real-world applications. The third compared framework, CNN U-Net, could only operate on grid inputs/outputs, and we suggested possible remedies to extend it to work with arbitrary unstructured meshes. When looking at prediction capabilities, especially interesting are the capabilities of the Perceiver~IO network, which demonstrated to give better predictions with a lesser number of parameters, as compared to MAgNET and CNN U-Net. Additionally, the use of Perceiver~IO creates a direct link to rapidly advancing research in ML and AI communities, which promises further advancements.

One of the first possible future extensions of the presented frameworks would be to incorporate the physics-informed neural network paradigm. This can be easily achieved by incorporating relevant physical laws in the optimization objective of the training procedure. Such extension can further increase the accuracy of predictions and accelerate the training procedure. Another possible extension is to consider a much wider class of models, including history/time-dependent phenomena (visco-elasticity, dynamics, plasticity, etc.), which would allow tackling more challenging problems in solid mechanics. Going beyond mechanics, these approaches can be also adopted for a much wider range of engineering and scientific applications.

\section*{Acknowledgments}
\begin{wrapfigure}{l}{0.34\textwidth}
    \includegraphics[width=0.3\textwidth]{PDF_examples/flag.jpg}
\end{wrapfigure}

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 764644.
Jakub Lengiewicz would like to acknowledge the support from EU Horizon 2020 Marie Sklodowska Curie Individual Fellowship \emph{MOrPhEM} under Grant 800150.
This paper only contains the author's views and the Research Executive Agency and the Commission are not responsible for any use that may be made of the information it contains.

Stephane Bordas, Jakub Lengiewicz and Ra\'ul I. Sosa are grateful for the support of the Fonds National de la Recherche Luxembourg FNR grant QuaC C20/MS/14782078. Stephane Bordas received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 811099 TWINNING Project DRIVEN for the University
of Luxembourg.

\bibliographystyle{Frontiers-Harvard}

redpajama_set_name: RedPajamaArXiv


text:
5 artists on how masculinity informs their practice By Clara Malley
5 artists on how masculinity informs their practice
Clara Malley
As gender moves beyond the binary, Document takes a look at toxic and beautiful permutations of masculinity with Liana Finck, Rashaad Newsome, Cédric Rivrain, Gianni Lee, and Amber Vittoria.
Underlying the debates over reproductive rights, the right to healthcare for queer individuals, or the latest misogynic soundbite hurled by the 45th president, lies the thorny subject of masculinity. Whether termed "toxic," "traditional," or "violent," the topic rears its head in controversies across industries, from journalism and politics to fashion and the arts, often involving the transgressions of male industry leaders. But even as individual stories move in and out of the spotlight, the thorny subject of masculinity remains at play.
The recent release of the American Psychological Association's "Guidelines for Psychological Practice with Boys and Men" elevates the subject to a public health concern. The guide urges medical professionals to consider the consequences of internalized masculine social norms when treating patients, particularly in situations involving mental health. The guide triggered various responses, some praising the proactivity of these measures while others criticized the implication that aspects of masculinity might be fundamentally harmful.
With this conversation in the periphery, Document asked four artists to comment upon their experience as part of an ongoing series exploring toxic, traditional, and beautiful permutations of masculinity. As society and culture evolve beyond the gender binary, their artistic practices navigate and challenge long-held beliefs about gender performance, violence, and self expression and the work still undone.
Drawings by Liana Finck. © Liana Finck.
Clara—Do you believe masculinity can be toxic? If so, how?
Liana Finck—I believe treating humans as "women"—in other words, as people whose only job is to be pleasant and nurturing—can be toxic. Men do it; women do it, too.
Clara—How does your work deconstruct this understanding — if you feel it does?
Liana—I did write a chapter of my graphic memoir about my dad—how it was hard for him to admit his vulnerabilities to himself or others because so many people relied on him. I think the patriarchy often forces men to hide their weakness, the same way it forces women to hide their strength.
Clara—Do you see interpretations of masculinity shifting in your industry as interpretations of femininity and queerness evolve?
Liana—I do think it's a hard time to be a "straight cis white" man. Women and other persecuted groups are asserting their needs these days, and the job of "straight cis white" men is to listen contritely. There's a lot of shame going around. On the other hand, it must be nice to have brain space for any type of self-expression that isn't about identity and anger.
Left: Look Back at It, 2016. Right: Through the Looking Glass. © Rashaad Newsome.
Clara—What comes to mind when you think of masculinity?
Rashaad Newsome—A product of patriarchy that blocks people from rich forms of intimacy and perpetuates an irrational loss of their patriarchal place in society.
Rashaad—In an imperialist, white supremacist, capitalist, patriarchy I don't really see how masculinity can ever be positive. Its very core is rooted in the divide of "genders." This divide supports patriarchal ideals, which breads homophobia, white supremacy, imperialism, and capitalism, which are all structures we need to work on freeing ourselves from.
Clara—How does your work engage with traditionally defined masculinity—strength, stoicism, perhaps violence?
Rashaad—I don't think that "traditionally defined masculinity" is strong or stoic but it can be quite violent. I think my work engages with it by using images from popular culture that participate in "traditionally defined masculinity," which traffics in the culture of domination. Through composition and juxtaposition, I try to create abstract images that interrogate masculinity and all of its cohorts that collaborate in the creation of systems of oppression that affect all people.
Left: Nassim, 2018. Right: Rodrigue, 2018. © Cédric Rivrain.
Cédric Rivrain—I think of a received idea: a historical and social construction. One can never think enough that Simone de Beauvoir's sentence, so often quoted and commented on, "One is not born, but rather becomes, a woman", implies its masculine corollary and supposes that the man also results from this construction.
Cédric—I am interested to circumvent it, to propose variations. Most of my models question themselves, in their life, these gendered partitions. I feel close to them, so I paint them very naturally. If I have occasionally portrayed men whose image corresponds a priori to the traditional idea of masculinity, it has always been by cracking this image, being primarily attentive to their emotion. This trouble is anyway already induced by my gaze on them, my painter gesture inseparable from my queer sensibility. It seems to me that in my portraits the qualities that we generally lend to the masculine, the ones you quote, for example, are rather assumed by women and vice versa.
Cédric—The men I draw, whom I paint, whether they are posing friends or figures I invent, always expose the most sensitive, fragile, certain would say feminine part, of masculinity. They can be crying, or on the verge of tears or having a vague look. The bodies, when they are naked, are not sexualized and especially not in a search of power or hardness but rather of abandonment to oneself, to the other, a form of absence which is never associated spontaneously with manliness. During my last personal exhibition in Paris, some portraits of nudes were visible from the street through the windows. It was very interesting to see the reactions of people, mostly men, of all ages. They were rather violent reactions, hostile or shocked, precisely because these men were not used to seeing male bodies naked, sensual without being sexual, represented with softness, tenderness and emotion. I believe that this is at the heart of the problem: dissident, marginal masculinities, are not sufficiently represented in the public space.
Left to right: I'm On My Deen; Self destruction by the hands of a glock; Roman Empire. © Gianni Lee.
Gianni Lee—When I think of masculinity, I think of the crossroads of expression and being. It's far beyond the characteristics of men that define masculinity. It's more so a language that we all speak; with any language, you have good words and bad words.
Gianni—Australian sociologist Raewyn Connell did a study on hegemonic masculinity where she argues it's use of "toxic" practices, such as physical violence, which may serve to reinforce men's dominance over women in Western societies.
As with all "dominant" ideals, the one practicing will always experience a sort of privilege. Being a man, I've experienced that privilege. I may refuse to see the errors in my thinking because of my privilege and upbringing. Of course, through my study of art, I have learned and shed some walls that the toxic parts of masculinity has built up over the years.
Gianni—As interpretations of femininity and queerness evolve, I feel as a heterosexual male I am becoming an endangered species. While I support interpretations of masculinity shifting, I am a strong believer in some traditional forms of masculinity. I feel all variables should be explored to create a more well rounded multi-dimensional man rather than killing off everything in favor of the new ideals. Social media has a way of sensationalizing this new form of man without actually showing people that they have options in manhood. In other words I find nothing wrong with rough housing , brotherhood, tough love ,sporting institutions and other classic structuring of manhood.
On Your Mark. © Amber Vittoria.
Amber Vittoria—Ideally, masculinity and femininity live in a balance within all of us; masculinity should be defined differently on a person-to-person basis, as the combination of our individual traits are distinctive. My masculine elements are honest reflection of my actions and strength in my range of emotions.
Amber—My work focuses on the idea of femininity and breaking down its societal tropes. The aim with my work is to reveal how femininity and masculinity live in harmony and to dismantle the stereotypes associated with femininity.
Amber—By leveraging physical traits such as body hair, overtly extended limbs, and rounded features, my pieces allow the viewer to define what is feminine within themselves.
Related: 5 Photographers on how masculinity informs their practice
Amber VittoriaArtCédric RivrainDrawingsGianni LeeIllustrationLiana FinckMasculinityPaintingRashaad Newsome
Photographer Danny Lowe captures generations of gypsy fashion at Appleby Horse Fair

redpajama_set_name: RedPajamaCommonCrawl


hi we were asked by a family member to rehome his shih tzu last year as he didny have time for her any more and it was the best thing we ever did. princess is 2 years old and is grey and white.
our family consists of me, my husband and out two girls aged 11 and 15 and our little princess of course.
she is not kc registered but we dont mind. she is so loving and follows me everywhere and i even find her curled up to me when i wake up on the morning if i dont close my bedroom door properly.lol.
we had to have her clipped very short when we got her as she was very matted and have only just found a groomed that we trust with her, so we are hoping to grow her hair a little now.
we are now looking for a little boy to complete our family, but only if he gets on with our little princess.
Hi teamfoster and welcome to the forum. It's great to have you here. We are a very friendly bunch of Shih Tzu addicts who love our fur babies to bits.
You have done a lovely thing to re-home princess. She obviously loves you to bits, and is very settled with you. Have a good look round the forum and you will find loads of advice, if you have any specific questions just ask away and someone will soon be along to help.
and welcome from me and ruby too.
its the best thing we done was find this site,there is loads of advice and help with everything you need to know about these beautiful dogs.
» Lots of rain and no sun - advice?


Easter is now around the corner (late this year!) and I have no doubt that many will be looking forward to the break and all it offers. Time with family, fun, friends and Easter eggs, especially with most schools ending their terms for their rest in the weeks before Easter, rather than after.
We have also moved into the new tax year 2019/2020 (starting 06 April 2019) and the Spring Statement from The Chancellor, Philip Hammond MP, on 13 March 2019 did not add much to the expected plans and allowances for the tax year ahead, although considering the political turmoil of recent months, it will be interesting to see if the next full Budget (usually around November each year) is brought forward.
Nature comes alive at this time of year as the draw of the great outdoors begins, with a few buttered hot cross buns and coffee to help the sunshine along. However, as the winter ebbs away, now might be a great time to think about using your annually renewed tax allowances early in the tax year, so that you can enjoy the summer without any concern about missing any allowances as the tax year progresses.
These are only some of the examples of allowances that can be used, noting that the nil rate tax bands for savings interest (£1,000 gross in a tax year for basic rate tax payers) and dividend income (£2,000 gross in a tax year) remain unchanged.
We appreciate (and hope) that you have many more exciting things to look forward to this coming Easter. Whatever your plans, we hope you have a relaxing and enjoyable time.
Please do contact the team at Chapters Financial in Guildford to consider your needs in this new tax year. No individual advice is provided during the course of this blog.


text:
A pedestrian walks past Yankee Stadium as it remains closed due to COVID-19 concerns, Thursday, March 26, 2020, in the Bronx borough of New York. To baseball fans, opening day is an annual rite of spring that evokes great anticipation and warm memories. This year's season was scheduled to begin Thursday, but there will be no games for a while because of the coronavirus outbreak. Major League Baseball has postponed opening day until mid-May at the earliest. (AP Photo/John Minchillo)
Yankee Stadium Is Turning Into A Giant Drive-In Movie Theater This Summer
There will also be games, raffles and a car-side dinner service at the screenings!
May 22, 2020 October 2, 2021 by Marie Rossiter
An organization in New York City is working on an out-of-the-box idea to bring life back to one of the city's most iconic gathering places that's been sitting eerily empty for months.
With no Major League Baseball games scheduled to be played in the foreseeable future, MASC Hospitality Group, a New York-based events organizer, saw all the empty parking spaces at Yankee Stadium and saw the opportunity to create what it calls a "social distancing playground."
The organization decided to take advantage of the recent resurgence of drive-in movies and announced the Uptown Drive-In Experience.
Organizers plan to offer a "complete drive-in experience with live performances from local artists, car-side dinner service from our fantastic street vendors, drinks from local and national brands, a movie feature presentation, games, raffles and tons of giveaways," according to the event's official website.
Friday, Saturday and Sunday nights are intended for more of a date-night feel, with a more family-friendly brunch series scheduled for earlier hours on Saturday and Sunday.
The Uptown Drive-In Experience is one of a number of community activities being planned by MASC Hospitality this summer for New York City, which was hit especially hard by the coronavirus pandemic.
A protective mask hands from a guests rearview mirror as "Trolls World Tour" is shown at the Four Brothers Drive In Theatre, Friday, May 15, 2020, in Amenia, N.Y., during the coronavirus pandemic. (AP Photo/John Minchillo)
A protective mask hands from a guests rearview mirror as "Trolls World Tour" is shown at the Four Brothers Drive In Theatre, Friday, May 15, 2020, in Amenia, N.Y., during the coronavirus pandemic. (AP Photo/John Minchillo)
"Since we were planning on having the Bronx Night Market, the Bronx Beer Festival and Uptown Epicure Festival running this summer, which was supposed to start in May, we already had a lot of great options available," Marco Shalma of MASC Hospitality Group told Time Out. "Everyone is so excited about the possibility of connecting sooner [at this new experience] than initially planned."
Tickets will be available to buy online and a waiting list is currently available on the event's website. The price of the all-inclusive movie ticket has not been released yet but Shalma hopes that information will be available soon.
"We're working with city and state officials, and some of our sponsors to reduce cost as much as we can to be able to make this very affordable, and for as many New Yorkers as possible," Shalma told Time Out. "Also, with the help of our partners, we will be able to raffle a bunch of tickets."
Categories Entertainment, Holiday & Seasonal, Movies & TV, News


The Puako General Store has been a feature in Puako since 1988. We offer freshly made food, groceries, cold drinks, beach items, delivery and more.
Our goal is to provide a convenient location that caters both to the local community and its visitors. We create an atmosphere where people can gather on the lanai and drink their morning coffee while reading the newspaper and enjoying the ocean breeze.
We have a variety of locally crafted foods & products as well as all your basic needs. Come in to shop, or have it delivered to your door.
Choose from a selection of fresh sandwiches, wraps, salads and poke made from ingredients purchased locally whenever possible.
The Puako General Store is run by an amazing team of friends and family members that love to spread the ALOHA spirit to everyone that comes in to see them!


text:
Bon Jovi, Jimmy Kimmel, Chris Christie Attend Howard Stern's 60th Birthday Bash
Feb 1, 2014 at 12:01 AM Feb 1, 2014 at 1:25 PM
Howard Stern celebrated his 60th birthday in style Friday night.
Celebrities including Bon Jovi, Joan Rivers and Jimmy Kimmel who emceed the event at New York City's Hammerstein Ballroom were on hand to offer their best wishes to the ...
Celebrities including Bon Jovi, Joan Rivers and Jimmy Kimmel - who emceed the event at New York City's Hammerstein Ballroom - were on hand to offer their best wishes to the radio host, the New York Post reports.
Stern broadcast the bash on his radio show and tweeted afterwards that the celebration was "perfect."
How to lose the Super Bowl: The worst plays in the game's history
Embattled New Jersey Governor Chris Christie was also in attendance. After reminding the crowd that Sunday's Super Bowl was taking place in his state, Christie introduced performers Bon Jovi.
Kimmel provided some good-natured ribbing of Stern, joking that he's "the most profoundly self-obsessed person on the planet," according to the Post.
Other attendees included Steven Tyler, Rosie O'Donnell, Robert Downey Jr. and Tracy Morgan. John Fogerty, The Black Keys and Adam Levine also performed.
On Saturday morning, Stern noted via Twitter that he was "still recovering."
View original Bon Jovi, Jimmy Kimmel, Chris Christie Attend Howard Stern's 60th Birthday Bash at TVGuide.com
Other Links From TVGuide.com Howard SternJoan RiversRosie O'DonnellTracy MorganJimmy KimmelBon JoviJohn FogertySteven TylerRobert Downey Jr.Adam LevineThe Black KeysChris Christie


text:
Gbenga Akinnagbe: Bio, Height, Weight, Age, Measurements
Actors, Author
Gbenga Akinnagbe is an American actor and writer. He is best known for his portrayal of Chris Partlow on the HBO series The Wire, and for his role as Larry Brown on the HBO series The Deuce. He has also appeared in films such as The Savages, Edge of Darkness, Knucklehead, Lottery Ticket, Fort Bliss, Detour, Crown Heights, Egg, The Sun Is Also a Star, and Render to Caesar, and the television series Barbershop, The Good Wife, Nurse Jackie, A Gifted Man, Damages, Law & Order: Special Victims Unit, Graceland, 24: Live Another Day, The Following, and Limitless. His two articles was published in The New York Times, one detailing a trip to Nepal to climb the Himalayas, and the other outlining the medical procedures he underwent to correct his severely flat feet. Born on December 12, 1978 in Washington, District of Columbia, USA to Nigerian parents, he was raised in Silver Spring, Maryland. He is the second oldest of six children, with one older sister and four younger brothers. He attended Colonel Zadok A. Magruder High School in Rockville, Maryland and graduated from Bucknell University with a degree in Political Science and English in 2000.
Gbenga Akinnagbe Personal Details:
Date Of Birth: 12 December 1978
Birth Place: Washington, D.C., USA
Birth Name: Gbenga Akinnagbe
Nickname: Gbenga
Also known as: Enitan Temitope
Occupation: Actor, Writer
Race/Ethnicity: Black
Gbenga Akinnagbe Body Statistics:
Height in Feet: 5′ 10″
Gbenga Akinnagbe Family Details:
Spouse/Wife: *
Children: *
Siblings: He has one older sister and four younger brothers.
Others: Wale Folarin (Cousin)
Gbenga Akinnagbe Education:
Colonel Zadok A. Magruder High School, Rockville, Maryland
Gbenga Akinnagbe Facts:
*He was born on December 12, 1978 in Washington, District of Columbia, USA.
*He attended Bucknell University on a wrestling scholarship, where he majored in Political Science and English.
*He was a star wrestler in college.
*He speaks Spanish fluently.
*His cousin is DC rapper Wale (Olubowale Victor Akintimehin).
*Follow him on Twitter and Instagram.


"Two potato chips are in a relationdip"—a pun on "relationship"—was posted on Twitter on April 16, 2013.
Q: What do you call it when 2 chips fall in love?
Two potato chips are in a relationdip.
What do you call it when two chips fall in love?
What do you call it when two chips fall in love????
What do you Call When Two Chips Fall in Love?


The basics – What is Pinterest?
Men are too, but let's face it. It's women. 200 million people use Pinterest every month, 70% are women.
What are women looking at on Pinterest?
Recipes. Fashion. Workout routines. Tattoos. Summer activities for kids. Vacation destinations. Organization tips. Cleaning products. The best pillows for a good night sleep. Wood pallet life hacks. Margarita recipes.
That is just my Pinterest search behavior from last week!
My point is, people use Pinterest similarly to Google and YouTube. It is more of a "search engine" than it is social media platform. Users are able to follow people, brands, boards (topics) that are interesting to them, but users don't necessarily follow friends, like they do on other social platforms.
If your business is utilizing digital media already, I recommend adding Pinterest into that mix. Google Search is still, and will be for the foreseeable future, the primary digital platform to connect with in-market buyers. Be there first. But adding Pinterest to that mix is a good idea. Pinterest is a great location to showcase your business because the people on Pinterest are looking for something… maybe that something is your product.
To walk through this process I am going to use a real world shopping scenario: running shoes. Let's say you sell shoes. And you want to sell more shoes to people who live in your area or shop on your ecommerce website. After you create a Pinterest Business page, you can "create new pin." There are two ways to do this – upload an image or Pin from a URL.
If when you create your pin, you choose to grab an image that already exists on your website, the alt-text from your image will pull the content for your pin. You can edit the text to make changes, but basically, the details of the pin are pulled from the content that does (or should) already exist on your website.
Use hashtags and relevant keywords – Pinners will find your pins.
Pinterest also provides an opportunity for business accounts to advertise their pins to further their reach. There are two ways to promote a pin. One, just select the red Promote button.
This is a quick promote option and allows you to set your budget, timeframe, and the keywords you want your pin to be populated for. Simple. Done. But, if you have a regional business, having your pin advertised to ANYONE on pinterest maybe isn't the right method.
In the Pinterest Ads Dashboard you have a lot more targeting options.
Similar to other social platforms, Pinterest gives you an audience size so you can help determine how many pinners match that targeting.
View your stats and results.
Pinterest gives you an ad dashboard you can use to monitor your ads and view your stats data. As a Business page, you can also access Analytics. Analytics data will show you more about who is viewing your pins and participating with your brand. You will also be able to see what other interests your followers/audience has on Pinterest.
You will also have impression and click data from your pins and brand as well.
Pinterest is awesome. You should be there. We can help you do it.
Shopping starts on Pinterest. Before a shopper knows exactly what they want to buy, they do some research and look for ideas. This research phase often happens on Pinterest. A poll of pinners recently conducted by Pinterest indicates that 72% of pinners were inspired to shop when they weren't initially looking for anything particular, based on pins they saw. In the same pole, 90% of pinners say Pinterest helped them decide what to purchase.
If Pinterest isn't part of your current advertising plan, it probably should be. If you need help, we can do that. We can also swap a few margarita recipes, too.
Emily Sorenson is the Digital Marketing Director at HenkinSchultz. She spends most of her time knee deep in analytics, data and stats. Her Pinterest boards mostly compose of recipes that may or may not even be made. Digital marketing is comprised of many different tools and tactics – Emily enjoys working with clients to develop digital strategies that work for their business and drive value.
For more see Emily's recent blog about the ABC's of digital marketing.


Many countries with relatively low birth rates have introduced incentives such as extra payments and other benefits for mothers.
Russia's birth rate has been declining for decades: the population fell from 149m in 1991 to 140m in 2018, and the median age has risen from 33 to 39.
Serbia, one of Hungary's neighbours, has one of the world's fastest-shrinking populations. It has seven million people and a median age of 43.
The birth rate in Italy is among Europe's lowest, along with Cyprus and Spain. Italy gives mothers an allowance of €80 per month (£70; $90) for each child born. The poorest families get a monthly allowance of €160 per child.
Identify and note the issues covered in this video.
Failure to do so will turn this dividend into an unbearable and lasting economic and social cost.


The Mercedes-Benz C-class has always been a fine entry luxury sports sedan, but it must gall the front office in Stuttgart that it has never had the sporting appeal of its cross-country rival from Munich, the BMW 3-series.
Mercedes continues to work on the problem. It now stresses improved dynamics of the '08 W204-model C-class, having added a new adaptive damping system and employed a longer and wider stance. This latest C-class has 1.8 more inches between the wheel centers, 108.7 inches overall, and a 1.1-to-1.4-inch-wider front and 2.4-to-2.7-wider rear track, depending on engine. Inside, there are now 1.6 more inches of front shoulder room.
Externally, the fenders of the C-class bulge something like the current S-class's, a distinction that is more apparent in the flesh than in photos. And there is now some visual differentiation between the Sport and Luxury models. Sport models have the three-pointed star mounted in the grille itself; Luxury versions have the traditional badge standing proud on the hood. Despite the size increase, Mercedes says the C-class weighs about the same as its predecessor - in the C350's case, 3550 pounds compared with 3515 pounds for a C350 Sport we tested in November 2005. Credit goes to the use of high-strength steel for 70 percent of the body shell. The car's weight distribution is also said to be improved, from the previous 54 percent up front and 46 percent in back to 52/48, due in part to mounting the engine lower and farther back in the chassis.
The C-class adopts a few features from the S-class to give a more premium interior feel. The iDrive-style controller in the center console operates the pop-up 4.9-inch-wide screen for the infotainment system. The instrument cluster, too, apes that of the S-class, with a screen instead of conventional gauges.
There will be two models initially in the U.S.: the C350 Sport, which comes standard with a seven-speed automatic transmission, and the C300, in Sport and Luxury versions. The C300 Sport will have an optional six-speed manual. The cars use the existing 268-hp, 3.5-liter V-6 and 228-hp, 3.0-liter V-6, up 10 and 7 horsepower, respectively.
In northern Sweden, we got a chance to find out whether Mercedes' increased emphasis on sporty handling has been translated into reality. "Our target was to develop long-distance-driving comfort and accentuate it with sportiness," says Klaus-Peter Claar, head of C-class development.
The C-class suspension layout is carried over - a strut and control-arm arrangement at the front and a multilink setup at the rear - with revised geometry, bushings, and subframes. One main element of the suspension is amplitude-dependent damping, which Mercedes calls "agility control." It's not electronic - it uses hydromechanical shock absorbers with an extra chamber to which oil is diverted by a control piston reacting to low road impulses. So in easy-driving situations, the damping force is reduced. Apart from preserving a comfortable highway ride, this system reduces the body's roll angle by up to 10 percent in some maneuvers compared with the old car. A new steering rack transmits more road feel, and the upshot of all the changes is a more stable car that has excellent body control. Sport models have a suspension that is lowered by 0.6 inch and features stiffer springs and dampers.
At a later date, Mercedes will offer the Advanced Agility package, which turns the car into a serious sports sedan. That package includes a variable damper system. Seven sensors provide electronic data that enable continuously variable damping at each wheel, and there is a choice, via a button alongside the gear selector, of comfort and sport programs. In addition to altering the damping, selecting "sport" changes the automatic-transmission shift strategy and accelerator characteristics. The Advanced Agility package also has more direct steering, with a ratio of 13.5:1 compared with the standard car's 14.5:1.
Our time with the C-class made it clear that this is a much sportier car, but whether it's enough to upset the BMW 3-series will have to wait for a comparison test. There's no word yet on whether Mercedes will bring a diesel-engined version to the U.S. The car goes on sale late this summer.


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How to exercise in the heat
7 simple tips for enjoying a safe outdoor workout this summer.
When exercising in hot weather, there's one guarantee: you will sweat - a lot!
While getting active outside makes a nice change from at home workouts, there are a few precautions to keep in mind as the temperatures start to rise. "Heat has a major effect on our fitness capacity," explains Ashleigh Kast, a New York City–based trainer and founder of Sophisticated Strength.
The best way to avoid potential pitfalls - like sunburn, dehydration and heat exhaustion - is to be prepared. Wearing the correct clothing, staying hydrated, and working out during optimal hours will help ensure you have a successful workout.
Here, three fitness professionals share the most important things you need to know to move safely in the heat:
1. Rise, shine, and get going.
Temperatures tend to be more bearable first thing in the morning or later in the day when the sun isn't directly overhead, explains Michelle Lovitt, an exercise physiologist and trainer in Los Angeles. No matter what time you head outside, always wear sunscreen, she adds. If you have to work out during the middle of the day, consider shortening the length of your workout to avoid overexerting yourself in the hot weather.
2. Start planning the night before.
Staying properly hydrated is always important, and it's extra important when you're exercising in the heat. "Freeze a water bottle the night before and carry it with you," says Lovitt. "It will keep you cool and hydrated during your workout, especially as the ice melts."
3. Keep tabs on the humidity.
It's not just the temperature that matters: relative humidity plays a big role in how hot it feels during your outdoor workout, explains Lovitt. "Humidity impacts the ability of the body to cool itself off, because sweat doesn't evaporate from the skin as quickly. As a result, the temperature feels much hotter than it actually reads."
4. Wear the right sneakers to avoid blisters.
When the temperature's climbing and you're sweating (everywhere), blisters are a major concern, and a major workout buzzkill. "Wear a shoe that breathes and fits just right," explains Kast. In addition to the fit, take the material into consideration. Look for airy mesh or a light fabric and avoid leather sneakers if possible, she adds.
5. Know the signs of dehydration.
It's natural to feel sweaty and tired when you're exercising. It's called "working out" for a reason! However, when you're exercising in the heat, it's important to be diligent about looking for symptoms of exhaustion or dehydration. Feeling nauseated, experiencing dry mouth, losing colour in your skin, and feeling confused, dizzy, or faint are potential warning signs explains Lindsey Clayton, a New York City–based trainer and co-founder of the Brave Body Project. "If you're excessively sweating, light-headed, or nauseated, you should complete your workout indoors or shorten it," she says.
6. Use smarter intervals.
Instead of heading outside and logging high-mileage walks, jogs or cycles, aim for a shorter workout of speed-walking intervals or tackling a few incline hills. If you're doing an interval workout of walking bursts followed by recovery periods, extend your rest breaks longer than normal. Kast suggests keeping a 1:1 work-to-rest ratio or one with a higher rest period.
7. Dress the part.
Your inclination may be to go as bare as possible, but a shirt that wicks away sweat can be a great makeshift towel for wiping sweat or dirt off your face, and may ultimately be a better choice, explains Kast.
Remember to discuss with your GP whether it's safe for you to exercise in hot temperatures.
Ready to get going?
Join WW today


We often don't pay much attention to our feet, even though we use them to get around. Feet can become susceptible to injuries and diseases, especially when we age. Feet lose cushioning, skin becomes dry, and nails become brittle. On top of this, many seniors suffer from poor circulation, which can make it difficult for foot sores to heal. Fortunately, caring for feet is not too difficult. By taking some precautions, you can promote good health and help your elderly patient prevent foot issues.
Once a foot problem develops, providing proper care can be time-consuming and costly. Issues with the feet can be prevented with regular medical checkups. As a caregiver, you should make sure your patient has a foot exam at least once a year. If you have diabetes, you may need to see a podiatrist more often.
Seniors are prone to several foot conditions. Dry skin can cause itching and cracking. Use moisturizers – particularly those that contain petroleum jelly – to keep feet hydrated. Just avoid putting lotion between the toes.
Athlete's foot is also common. Despite the name, anyone can get this condition – not just athletes. This fungal infection thrives in dark and wet conditions. Redness, peeling, itching, and blisters are common symptoms. Fortunately, there are many sprays and ointments that can help treat this condition. You can prevent this by keeping feet clean and dry.
Ingrown toenails can be painful. Trim the nail straight across and keep it even with the top of the toe. Do not attempt to pull the nail, as this can cause infection.
Corns and calluses can occur when you wear tight-fitting shoes. If your patient has these, you can use a pumice stone to file them down. Just be wary of using medicated pads designed for corns and calluses. These can harm healthy skin and make matters worse.
Too much pressure on the feet can cause heel spurs. These are calcium deposits that form on the heel. To alleviate the pain, use heel pads and other forms of support.
The best way to prevent foot issues is to protect them. While tight shoes like high heels may be stylish, they lack comfort. This pressure can cause corns and calluses. In addition, feet tend to get wider as you age, so make sure your patient is wearing comfortable shoes and socks.
Exercise is essential. Your patient doesn't need to work out at a gym or run a marathon, but he or she should engage in some light activity. Walking and stretches are great ways to prevent the feet from stiffening up.
Your patient can also protect his or her feet by avoiding smoking, which affects circulation by narrowing the arteries.
According to the Centers for Disease Control and Prevention, more than 29 million people in the United States suffer from diabetes. Diabetes is a chronic condition that can affect the entire body – including the feet. A minor issue can quickly turn into a serious problem. That's because diabetes damages blood vessels, which can cause wounds to heal slowly. A small cut or blister can lead to gangrene, which may require amputation of the foot. To further complicate the issue, diabetes can make you unable to feel pain, so many patients have wounds and foot diseases and don't even know it.
If you have diabetes, you can prevent foot problems by keeping your blood sugar under control. Clean feet are healthy feet, so wash them in warm water daily. Check them daily for any blisters, calluses, or corns. Do not try to remove them yourself, as you could cause damage to your feet. Avoid going barefoot, as you could step on something that could cause injury to your feet. Always wear shoes, or socks at a minimum. Keep the blood flowing. Try to move around and exercise. Even wiggling your feet is helpful.
While caregiving involves helping patients with basic functions such as eating, dressing, and grooming, it also involves disease prevention. Not only should you check your patient's feet regularly, but, if a cut is present on the foot, you may need to provide injury care for cleaning and dressing wounds. This will help avoid infections and other complications.


This blend is designed to support healthy immune functioning and may be especially helpful during winter when you want extra ammunition against common ills and chills. It assists the body's natural processes of recovery to help you get you back to health safely and quickly. Suitable for diffusion, inhalation or bath.
The Everyday Organics Infant Massage Oil by BUDS is made from Sunflower and Olive Oil is nothing but perfect for your baby and can be used whenever your baby wishes it.


Incorporate effortlessly cool Current/Elliott jeans into your staple wardrobe. Designed with a cropped, straight leg, the comfortable stretch denim is finished with trend-led raw hems for an authentic, worn-in look. Complete with belt loops, five pockets and logo hardware, they make the perfect foundation for casual looks.


Images and descriptions are for reference purposes only & may change without prior notice. PLEASE RELY ON ABOVE DESCRIPTIONS AND NOT ON IMAGES. NBDE460-BCBC Dell Urban Backpack 15" - Notebook Backpack Black is available from Dell Laptops and it is sold as new from our 15.4 / 15.6 inch notebook carry bags range of products. For reference purposes, the price of R576 is valid 2019-04-20, and while stocks last, may be valid until 2019-04-30 , but this is a guide only. Prices may change without prior notice. Final price is by quote only.T&C's apply. E&OE. Laptop Direct. You are looking at products found in the 15.4 / 15.6 inch notebook carry bags category.


text:
I Am Trying To Believe
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A t-shirt from the early 2007 European tour of Nine Inch Nails. The highlighted letters spelled out 'iamtryingtobelieve' which was found out to be a website when adding ".com" to it.
http://iamtryingtobelieve.com/
Number: 24.14.1 (T-Shirt) & 24.14.3 (Website)
Background Text: The Hobbit by J.R.R. Tolkien
2.1 Auto-Reply E-Mail
2.2 Unlinked Images
On February 12th, 2007, following the initial shows of the Nine Inch Nails 2007 European tour in Lisbon, Portugal, fans noticed a peculiarity on a tour T-shirt—several letters from the tour itinerary on the rear of the shirt were highlighted. The highlighted letters, in order, spelled out "iamtryingtobelieve," which, when adding ".com" to the end, was discovered to be an active website URL. Around the same time, the same URL was also linked to in an update on theninhotline.net.
The words "I am trying to believe" are also some of the main lyrics in the song "The Good Soldier."
Another reference could be found in the Year Zero trailer video that was posted on yearzero.nin.com. In the video, just before "The Presence" is seen, there is what appears to be a road sign that passes in front of the camera's view. If the video is played back in slow motion, one can see that the text on the sign is actually "I AM TRYING TO BELIEVE."
The content of the website centers around a drug called Parepin. The unnamed author of the website describes a future world in which Parepin has been introduced into municipal water supplies throughout the United States, under the direction of the US government. The purported reason for this put forward in the media as excerpted on the website is that Parepin is immune system booster that will protect Americans from biological warfare; apparently in response to biological terror attacks which have occurred on the west coast. However, the website author's main thesis is that Parepin has no immune system boosting effects, and is in fact a psychoactive drug being used by the government to sedate the population and make them complacent.
The site has several sections, describing what Parepin is, listing several undesirable side effects, and suggesting that people try to avoid Parepin. Also prominently featured is a mention of the phenomenon known as The Presence, which the author suggests is a Parepin-induced hallucination. On the bottom of the last page is an email address for contacting the author: water@iamtryingtobelieve.com
There also appears to be a mirror of this site set up at an alternative IP address, though DNS does not appear to link to it.
Auto-Reply E-Mail
Number: 24.14.2 (count consecutive dashes in first line of email, using underscore to seperate)
When sending an email to the address listed on the website, the following auto-reply is received:
------------------------AUTO_RESPONSE--------------_--
Thank you for your interest. It is now clear to me that Parepin is a completely safe and effective agent developed to protect us from bio-terrorism. The Administration is acting purely in the best interests of its citizens; to suggest otherwise was irresponsible and I deeply regret it.
I'm drinking the water. So should you.
Attempting to reply to this auto-reply produces no further result.
Unlinked Images
There have been 12 images found that are not linked in the HTML.
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