A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.

CONVERGENCE AND SUPERCONVERGENCE ANALYSIS OF LAGRANGE RECTANGULAR ELEMENTS WITH ANY ORDER ON ARBITRARY RECTANGULAR MESHES

This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensor-product families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.

Multi-Material ALE with AMR for Modeling Hot Plasmas and Cold Fragmenting Materials

We have developed a new 3D multi-physics multi-material code, ALE-AMR, which combines Arbitrary Lagrangian Eulerian （ALE） hydrodynamics with Adaptive Mesh Refinement （AMR） to connect the continuum to the microstructural regimes. The code is unique in its ability to model hot radiating plasmas and cold fragmenting solids. New numerical techniques were developed for many of the physics packages to work efficiently on a dynamically moving and adapting mesh. We use interface reconstruction based on volume fractions of the material components within mixed zones and reconstruct interfaces as needed. This interface reconstruction model is also used for void coalescence and fragmentation. A flexible strength/failure framework allows for pluggable material models, which may require material history arrays to determine the level of accumulated damage or the evolving yield stress in J2 plasticity models. For some applications laser rays are propagating through a virtual composite mesh consisting of the finest resolution representation of the modeled space. A new 2nd order accurate diffusion solver has been implemented for the thermal conduction and radiation transport packages. One application area is the modeling of laser/target effects including debris/shrapnel generation. Other application areas include warm dense matter, EUV lithography, and material wall interactions for fusion devices.

A NEW FREE SURFACE MESH TRACING METHOD

Free surface flow problems involving large free motions are analysed using finite element techniques. In solving these problems an Arbitrary Lagrangian-Eulerian(ALE)kinematical description of the fluid domain is adopted, in which the nodal points can be displaced independently of the fluid motion. A new mesh tracing method is proposed in this paper. To confirm the effectiveness of the new method, solitary wave propagation is analysed and the numerical results are compared with the analytical results. The behaviour of the viscous fluid flow with a free surface is expressed by the unsteady Navier-Stokes equation. For numerical integration in time the velocity correction fractional step method is used.