Hybrid Trefftz Finite Element Method And Thermal Response Analysis Of Heterogeneous Materials

It is urgent to understand the physical properties of new type composites during the structural design.Modern simulation technology,especially the finite element method,provides the possibility to achieve the goal.However,the conventional finite element method needs the mesh refinement in the vicinity of heterogeneity(inclusion)when modeling heterogeneous materials,which is time-consuming and results in high error rate.In order to eliminate the defects of conventional finite element method from the theoretical source,the hybrid Trefftz finite element method(HT-FEM)came into being.The Trefftz finite element formulation involves boundary integrals only,which provides unlimited number of sides and allows concave-convex shapes of constructed elements.This leads to great flexibility in mesh generation for the complicated geometry.At present,HT-FEM has been successfully applied to many fields such as heat transfer,elasticity and multi-field coupling.Especially,HT-FEM can achieve highly accurate results by adjusting the intra-element interpolation functions without mesh refinement when dealing with the structures with local effects such as inclusions,holes and cracks.Based on HT-FEM,many researchers have made great contributions to heterogeneous materials.The common approach in the literature is as follows: Firstly,the element domain(occupied by an inclusion with surrounding matrix)is divided into two boundary value subproblems;Next,the variational functional is established for each phase region;Then,the two functionals are linked by incorporating the continuity condition across the interface between two phases;Finally,the finite element formulation is derived.The approach mentioned above may be called a two-functional model.The boundary integral along the interface between phases is inevitable because the variational functionals have to be established for all individual phases.This increases the difficulty in the theoretical derivation and programming.Especially,it is very complicated or even impossible to establish multi-functional model when the interphases(such as coatings)are added between the inclusion and matrix.In order to overcome this limitation,a single-functional model is proposed to construct inclusion elements for the most widely used fiber-reinforced composites.In the framework of HT-FEM,a series of polygonal elements with a circular-,multilayer coatedand elliptical inclusion is constructed in combination with Voronoi tessellation technique.In the constructed element,a polygon is utilized to surround the inclusion,coatings and surrounding matrix,and a united variational functional(the single-functional model)is established for the whole polygonal element domain.The novel finite element formulation is obtained by applying the divergence theorem and performing the stationary of the united functional twice.The boundary integrals appeared in the novel formulation are performed along the outer-boundary of the matrix region only.All the intra-element interpolation functions involved in these boundary integrals depend directly on the T-complete solutions for the matrix region.The interaction between the inclusion,coating and matrix is transmitted through the interface continuity condition.Mathematically speaking,the multi-functional model introduces the interface continuity condition at the functional level while the single-functional model proposed in this paper is implemented at the intra-element interpolation function level.This avoids the boundary integrals along the interface deftly.In this paper,several numerical examples of heat conduction problems are selected to verify the effectiveness of HT-FEM.Compared with the results obtained by the commercial software ABAQUS,HT-FEM can still keep high accuracy and shorter computational time with coarse meshes.It is found that the composites exhibit anisotropy depending on the inclination angle when the elliptical fibers are arranged in the same direction.Additionally,based on single-functional model,the hybrid Trefftz finite element formulation is derived for the elastic inclusion problem.The construction of inclusion/coating/matrix element model can not only deepen the high-performance finite element theory academically but also provides novel idea for new type composite materials.