Hp‐Adaptive Finite Element Methods for Variational Inequalities

Advances in Adaptive Computational Methods in Mechanics

Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A parallel computing procedure for computing the bounds on the {\it J}-integral in functionally graded materials is presented based on a Neumann element a-posteriori error bound. The finite element solution of {\it J}-integral is first obtained on a coarser finite element mesh, then a-posteriori bounding procedure based on the finite element error estimate is used to compute the lower and upper bounds on the {\it J}-integral. The computation of the error estimate is performed by solving independent elemental Neumann sub problems decomposed from the finite element model, thus the computing procedure is parallel and potential to solve large scale structural problems. An example is given in the end of paper to compute the lower and upper bounds on the {\it J}-integral of functionally graded materials.

In many scientific problems, adaptive finite element methods has been widely used to improve the accuracy of numerical solutions. The general idea is to refine or adjust the mesh such that the errors are “equally” distributed over the computational mesh, with the aim of achieving a better accurate solution using an optimal number of degrees of freedom. By using the information from the approximated solution and the known data, the a posteriori error estimator provides the information about the size and the distribution of the error of the finite element approximation. There is a large numerical analysis literature on adaptive finite element methods, and various types of a posteriori estimates have been proposed for different problems, see e.g. [1]. The a posterior error estimate and adaptive finite element method were first introduced by [2]. Since the later 1980s, much research work on a posteriori error estimate has been developed including the residual type a posteriori error estimate [8], recovery type a posteriori error estimate [16], a posteriori error estimate based on hierarchic basis [5, 4], and so on. For the literature, the readers are referred to the books [1, 3, 12, 14], the papers [6, 13, 15], and the references cited therein.KeywordsError EstimatorPosteriori ErrorPosteriori Error EstimateInitial MeshElement Stiffness MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents recent work on developing methods for analyzing radiation heat transfer between diffuse-gray surfaces using p-version finite elements. The work was motivated by a thermal analysis of a High Speed Civil Transport (HSCT) wing structure which showed the importance of radiation heat transfer throughout the structure. The analysis also showed that refining the finite element mesh to accurately capture the temperature distribution on the internal structure led to very large meshes with unacceptably long execution times. Traditional methods for calculating surface-to-surface radiation are based on assumptions that are not appropriate for p-version finite elements. Two methods for determining internal radiation heat transfer are developed for one and two-dimensional p-version finite elements. In the first method, higher-order elements are divided into a number of sub-elements. Traditional methods are used to determine radiation heat flux along each sub-element and then mapped back to the parent element. In the second method, the radiation heat transfer equations are numerically integrated over the higher-order element. Comparisons with analytical solutions show that the integration scheme is generally more accurate than the sub-element method. Comparison to results from traditional finite elements shows that significant reduction in the number of elements in the mesh is possible using higher-order (p-version) finite elements.

This letter proposes to automate generation of reduced-order models used for accelerated S-parameter computation by applying a posteriori model error estimators. So far, a posteriori error estimators were used in Reduced Basis Method (RBM) and Proper Orthogonal Decomposition (POD) to select frequency points at which basis vectors are generated. This letter shows how a posteriori error estimators can be applied to automatically select the order of the reduced model in second-order Model Order Reduction (MOR) methods. Three different error estimators are investigated and compared in order to arrive at a new MOR scheme that is fast, reliable, and fully automated. The effectiveness of the proposed approach is verified by very high accuracy of the computed scattering parameters ( S-parameters) for an example of a waveguide filter over a prescribed frequency band.

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H1(Ω)‐norm, L2(Ω)‐norm and L2(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.

The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening.

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L2 norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L2 norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017

Boundary Integral Equations and A Posteriori Error Estimates

: The principal goals of the project are; The derivation of a uniformly stable boundary element formulations in the equations valid for the whole range of wave numbers k. The formulation should be compatible with the the C(O) regularity assumption for finite element approximations of the structure. The development of C(O) finite element approximations for one-, two-, and three- dimensional geometries. This part of the project included such fundamental issues as the development of new hp finite element data structures and constrained approximations for coupled boundary element/finite element problems. The development of a coupled finite element/boundary element code and the verification of expected rates of convergence. The derivation, implementation, and verification of an a posteriori error estimate for boundary element methods. The development of adaptive strategies. Having obtained a local error estimate, these strategies would provide for reducing mesh size or increasing spectral order so as to achieve a target error with a number of unknowns. The solution of selected problems. These are designed to demonstrate the effectiveness of the methodology on classical benchmark problems and, ultimately, on realistic scattering problems of interest in Naval research. The verification of the applicability of the hp approximation to structural mechanics, including bulky structures such as bulkheads, but also possibly shells, thin beams and plates. The development of a mathematical foundation for transient acoustics, including studying mathematical properties of linear hyperbolic systems associated with transient acoustics, the development of new transient hp-adaptive schemes, the development of high order transient solvers and associated a posteriori error estimates, and associated adaptive strategies.

In this paper we present a multiscale framework suited for geometrically nonlinear computations of foam-like materials applying high-order finite elements (p-FEM). This framework is based on a nested finite element analysis (FEA) on two scales, one nonlinear boundary value problem on the macroscale and k independent nonlinear boundary value problems on the microscale allowing for distributed computing. The two scales are coupled by a numerical projection and homogenization procedure. On the microscale the foam-like structures are discretized by high-order continuum-based finite elements, which are known to be very efficient and robust with respect to locking effects. In our numerical examples we will discuss in detail three characteristic test cases (simple shear, tension and bending). Special emphasis is placed on the material’s deformation-induced anisotropy and the macroscopic load-displacement behavior.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order 𝒪 ( H s Log ^ ( H h ) s 2 ) {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and 𝒪 ( H 2 ⁢ s Log ^ ( H h ) s ) {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the L ∞ ⁢ ( H 1 ) {L^{\infty}(H^{1})} -norm and L ∞ ⁢ ( L 2 ) {L^{\infty}(L^{2})} -norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.