where , , and . The latter definition provides convective heat transfer boundary conditions of the form: , (6) is satisfied for and . Note that the equality means , which gives , and in view of the identity leads to where the associated conductivity tensor on the tessellated domain is (7) for isotropic (or indeed, orthotropic) , gives rise to an orthotropic . This property arises from the fact that is evidently symmetric and positive definite, hence giving rise to at most three distinct positive conductivities in orthogonal directions. 3. THE TESSELLATED FINITE ELEMENT METHOD The finite element method can be derived in one of three ways: direct minimisation of a functional (e.g. minimisation of potential energy); A variational statement, provided one exists for the problem (e.g. Hamilton’s principle) and; the weighted residual method applied to the governing partial differential equations. A fourth approach has arisen in the literature [35], which avoids the use of the governing partial differential equation, provided a transport equation is available. The approach involves the direct weighting of the transport equation which, on application to Equation (4) gives , (8) where is a weighting function which in the first instance is assumed continuous at least up to the first derivative, i.e. and is independent of time, i.e.. A similar equation is assumed to exist for the tessellation (9) which, after substitution of Nanson’s identities and , , , and gives (10) which is identical to Equation (8) on setting . The finite element method applied to an element in the tessellation immediately follows, i.e. (11)
where , , and . The latter definition provides convective heat transfer boundary conditions of the form: , (6) is satisfied for and . Note that the equality means , which gives , and in view of the identity leads to where the associated conductivity tensor on the tessellated domain is (7) for isotropic (or indeed, orthotropic) , gives rise to an orthotropic . This property arises from the fact that is evidently symmetric and positive definite, hence giving rise to at most three distinct positive conductivities in orthogonal directions. 3. THE TESSELLATED FINITE ELEMENT METHOD The finite element method can be derived in one of three ways: direct minimisation of a functional (e.g. minimisation of potential energy); A variational statement, provided one exists for the problem (e.g. Hamilton’s principle) and; the weighted residual method applied to the governing partial differential equations. A fourth approach has arisen in the literature [35], which avoids the use of the governing partial differential equation, provided a transport equation is available. The approach involves the direct weighting of the transport equation which, on application to Equation (4) gives , (8) where is a weighting function which in the first instance is assumed continuous at least up to the first derivative, i.e. and is independent of time, i.e.. A similar equation is assumed to exist for the tessellation (9) which, after substitution of Nanson’s identities and , , , and gives (10) which is identical to Equation (8) on setting . The finite element method applied to an element in the tessellation immediately follows, i.e. (11)
CAS_paper_GALERKIN.docx