Today, computer aided engineering (CAE) has been used for supporting engineers in many tasks. For example, in a structure or product design procedure, CAE analysis, in particular finite element analysis (FEA), has often been employed to evaluate responses (e.g., stresses, displacements, etc.) under various loading conditions (e.g., static or dynamic).
FEA is a computerized method widely used in industry to simulate (i.e., model and solve) engineering problems relating to complex products or systems (e.g., cars, airplanes, etc.) such as three-dimensional non-linear structural design and analysis. FEA derives its name from the manner in which the geometry of the object under consideration is specified. The geometry is defined by elements and nodes. There are many types of elements, solid elements for volumes, shell or plate elements for surfaces and beam or truss elements for one-dimensional structure objects. Example elements are shown in FIG. 1.
With the advent of the modern digital computer, FEA has been implemented as FEA software. FEA software can be classified into two general types: implicit and explicit. Implicit FEA software uses an implicit equation solver to solve a system of coupled linear equations, while explicit FEA software does not solve coupled equations but explicitly solves for each unknown assuming them uncoupled. Additionally, to evaluate dynamic responses, either implicit or explicit FEA is conducted at a number of solution cycles or time steps in time-marching analysis. At each solution cycle a particular structural response is obtained. Time increment between two consequent solution cycles is referred to as time step Δt.
Impact events (e.g., car crash, metal forming, etc.) are preferably simulated using the explicit FEA. However, the explicit FEA method is numerically unstable, which can be overcome or compensated by using very small time step (in the order microsecond 10−6 second). As a result, huge number of time steps are required to simulate just a very short period of time (e.g., 0.1 second). Even with a state-of-the-art computer system, it takes a lot of real time (in the order of many hours) to perform this kind of analysis. Hence, it is more feasible to simulate static or quasi-static engineering problems using the implicit FEA, which allows a much larger time step size. Generally, the solution for each time step achieves convergence (based on a predefined tolerance) with one or more iterations.
One of the physical structural behaviors in simulating impact events is structural contact, which is defined as two portions of a FEA model touching each other in FEA analysis. One of the prior art solutions to simulate the structural contact is referred to as penalty method, in which a compensating force or spring is introduced after a structural contact penetration has been detected in a particular time step of a time-marching analysis. The size of the compensating force or spring is dependent upon the magnitude of contact penetrations.
The penalty method works quite well in explicit FEA. However, applying such method in the implicit FEA analysis, the simulation results are not as good and often not acceptable. In particular, the FEA mesh model would become distorted due to a large fictitious compensating forces or springs resulting from large contact penetrations between two time steps in the implicit FEA.
FIGS. 2A and 2B are diagrams showing implicit FEA results of a metal forming simulation at two consecutive time steps using a prior art approach. Shown in FIG. 2A, a first part or object (e.g., a blank sheet metal) 202 is pushed towards a second part or object 204 (e.g., a die configured to receive the blank) initially (i.e., step 0). At the next time step (i.e., step 1) shown in FIG. 2B, large contact penetration 220 results between the first and the second objects. It is evident that the simulation results are physically impossible. With such large contact penetrations, a correction measure based on the penalty method (e.g., applying counter forces or springs at the contact penetration locations) could not reverse damages in the FEA mesh model. Therefore, it would be desirable to have improved systems and methods of limiting contact penetration in numerical simulation of non-linear structure responses using implicit finite element analysis.