Weak Galerkin Finite Element Method For Linear Elasticity Problem

In this paper,the weak Galerkin finite element(WG)method is used to solve the linear elastic equation.The linear elastic equation is a classical model equation.Due to its complex structure and the fact that some of its physical quantities need to meet certain conditions,many difficulties are encountered in the numerical calculation of the linear elastic equation.Among the three issues that scholars are generally concerned about are:1.The speciality of the definition of strain tensor leads to the numerical format is difficult in satisfying the coercive property;2.The numerical solution of the linear elastic problem has a certain parameter dependence,and the parameter is unbounded when the elastic body tends to be incompressible;Stress tensor has symmetry due to Newton's third law,while the numerical format that maintains the stress tensor symmetry is very difficult to construct.This article uses the WG method to answer the above three questions separately.WG method is a generalization of the traditional finite element method.WG method use the shape regular polygonal or polyhedra partitions and it use the discontinuous piecewise polynomials to approximate the unknown functions,namely weak functions.Weak functions {u0,bb} have two parts,the inner part u0 and the boundary part ub.The boundary function ubis not necessarily related to the inner function u0,in the numerical algorithm,the bases are defined on both inner elements and the boundaries.For the weak function,we defined weak derivatives based on the integration by parts.Take the weak gradient for {u0,ub} as an example(?)Since the exact solution we considered is continuous,the weak continuity of the weak functions need to be considered.To this end,we introduce stabilizers.The stabilizer describe the differences of weak functions among elements.A typical form of stabilizer given by(?)It could be different for different problems.From the definition we can see the smaller the value of the stabilizer,the more continuous the function.Linear elasticity is a classical physical problem.It considers the relationship among stress,strain,and displacement in elastomer.Linear elasticity is a simplification of the general elasticity problem,it is difficult to get a solution.For example in the finite element method,the coerciveness of the scheme,the locking phenomenon,and the symmetry of the stress all these problems bring great difficulties to the analysis.This paper studies the application of WG method to the linear elasticity problem.Firstly,we consider the primal form of linear elasticity:(?)The WG algorithm is given and the well-posedness is derived.However we can't prove the scheme is locking-free directly.To this end,we define a corresponding mixed form:(?)The WG shceme of this mixed form is equivalent to the one for primal form and the mixed form is locking-free.As a result,the primal form is proved to be locking-free.The WG algorithm get the optimal convergence rate and numerical experiments verify the theoretical result.Except the displacement we always consider the stress.But compute stress by taking differences to displacement will loss the accuracy.So we consider the following mixed form:(?)(?)(?)(?)This mixed form is different from the mixed form given in the last paragraph,the last one is constructed corresponding to the primal WG scheme,and this one comes from the balanced function and the constitutive function of elastomer.It follows from the Newton's third law that stress is symmetric,which brings great difficulty to the construction of the approximate space of the finite element method.Even if there are some successful examples,but they are inevitably complicated.There are two way to overcome this obstacle.The first method is using weak symmetric stress and the other is using nonconforming elements.In this paper,we use WG method to compute the linear elasticity problem in mixed form.As a nonconforming element method,WG method use strong symmetric stress and holds for the coerciveness property and inf-sup condition.The WG method has optimal convergence order and numerical experiments verify the theoretical conclusions.WG scheme is easy to construct and applicable for many problems.But compared with the traditional finite element method,the degree of freedom for WG methods is larger.To this end,we consider the hybridized WG method.In the hybridized WG method,an auxiliary function is solved from a linear system where degree of freedom relates only to the element boundaries(?)The auxiliary function is called the Lagrange multiplier.It is the majority of the computing resource.Since the Lagrange multiplier defined only on the element boundaries,the computational costs could be reduced sharply comparing to the standard WG methods.