J. Tinsley Oden

An h-p adaptive method using clouds

Several computational and mathematical features of the h-p cloud method are demonstrated in this paper. We show how h, p and h-p adaptivity can be implemented in the h-p cloud method without traditional grid concepts typical of finite element methods. The mathematical derivation of an a posteriori error estimate for the h-p cloud method is also presented. Several numerical examples illustrate the main ideas of the method.

H-p Clouds - An h-p Meshless Method

A new methodology to build discrete models of boundary-value problems is presented. The h-pcloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomialreproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and a several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater -exibility than traditional h-p finite element methods. Several numerical experiments in I-D and 2-D are also presented. @ 1996 John Wiley & Sons, Inc. In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc. Moreover, in most computer-aided design work, the generation of an appropriate mesh constitutes, by far, the costliest portion of the computer-aided analysis of products and processes. These are among the reasons that interest in so-called meshless methods has grown rapidly in recent years. Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. The analysis of problems such as crack propagation, penetration, and large deformations, can, in principle, be greatly simplified by the use of meshless methods. A growing crack, for example, can be modeled by simply extending the free surfaces that correspond to the crack [ 11. The analysis of large deformation problems by, e.g., finite element methods, may require the continuous remeshing of the domain to avoid the breakdown of the calculation due to

The Mathematics of Finite Elements and Applications II

Abstract : This paper describes theory and methods for developing a posteriori error estimates and an adaptive strategy for hp-finite element approximations of the incompressible Navier-Stokes equations. For an error estimation, use is made of a new approach which is based on the work of Ainsworth, Wu and the author. That theory has been shown to produce good results for general elliptic systems and general hp-finite element method% Recently, these techniques have been extended to the Navier-Stokes equations A three-step adaptive algorithm is also described which produces reasonable hp meshes very efficiently. These techniques are applied to representative transient and steady state problems of incompressible viscous flow.

A unified approach to a posteriori error estimation using element residual methods

SummaryThis paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].

Solution of stochastic partial differential equations using Galerkin finite element techniques

This paper presents a framework for the construction of Galerkin approximations of elliptic boundary-value problems with stochastic input data. A variational formulation is developed which allows, among others, numerical treatment by the finite element method; a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized. The paper develops a foundation for treating stochastic partial differential equations (PDEs) which can be further developed in many directions.

A discontinuous hp finite element method for convection—diffusion problems

This paper presents a new method which exhibits the best features of both finite volume and finite element techniques. Special attention is given to the issues of conservation, flexible accuracy, and stability. The method is elementwise conservative, the order of polynomial approximation can be adjusted element by element, and the stability is not based on the introduction of artificial diffusion, but on the use of a very particular finite element formulation with discontinuous basis functions. The method supports h-, p-, and hp-approximations and can be applied to any type of meshes, including non-matching grids. A priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy.

A discontinuous hp finite element method for the Euler and Navier–Stokes equations

We introduce a new method for the solution of the Euler and Navier-Stokes equations, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver. The method involves a weak imposition of continuity conditions on the state variables and on inviscid and diffusive fluxes across interelement and domain boundaries. Within each element the field variables are approximated using polynomial expansions with local support; therefore, this method is particularly amenable to adaptive refinements and polynomial enrichment. Moreover, the order of spectral approximation on each element can be adaptively controlled according to the regularity of the solution. The particular formulation on which the method is based makes possible a consistent implementation of boundary conditions, and the approximate solutions are locally (elementwise) conservative. The results of numerical experiments for representative benchmarks suggest that the method is robust, capable of delivering high rates of convergence, and well suited to be implemented in parallel computers

Analysis and adaptive modeling of highly heterogeneous elastic structures

In this paper, we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly-elastic structures. The method, which represents a significant departure from traditional homogenization methods, provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures. An early variant of this method was first introduced in [IO]. The method, referred to as HDPM (Homogenized Dirichlet Projection Method), proceeds by first solving an auxiliary homogenized boundary value problem, describing the deformation of a structure with the same exterior geometry, but with a selected set of uniform material properties. The adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine-scale heterogeneous problem, without specific knowledge of this fine-scale solution. In those subdomains where the homogenized solution is deemed acceptable, it is retained. However, in subdomains where the homogenized solution is inadequate, as is determined when the estimated error exceeds a preset tolerance, a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains. These local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions. A posteriori error estimates are made to ascertain the quality of the resulting solution. If the quality of the solution remains inadequate, above a preset error tolerance, a two stage adaptive procedure is implemented. Stage-I (‘material adaptivity’) corresponds to modifying the homogenized structure’s material properties. If, after Stage I, the solution quality is still inadequate, the subdomains of local solution are enlarged, thereby modeling in greater detail the actual microstructure, and the local solution process is. repeated (Stage-II, ‘subdomain unrefinement’). The main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account. The cost of this method is shown to be orders of magnitude cheaper than direct huge-scale computational simulations of micromechanical events. The results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates. In this work, we present a method for the solution of boundary value problems modeling the deformation of structures composed of highly heterogeneous linearly-elastic materials. We concentrate on materials formed by a homogeneous matrix embedded with particulate matter of different properties (Fig. 1). However, the results presented are not limited to materials of this type, and can, in theory, be used for materials with virtually any microstructure. A straightforward calculation reveals that an accurate numerical approximation of fine-scale solutions of

Functionally graded material: a parametric study on thermal-stress characteristics using the Crank–Nicolson–Galerkin scheme

In this paper, we analyze thermal-stress characteristics of functionally graded materials (FGM), the newly introduced layered composite materials with great potential as next generation composites. Among the several material parameters governing its characteristics, we study the effects of the material variation through the thickness and the size of the FGM layer inserted between metal and ceramic layers using the finite element method. Through a representative model problem, we observe different thermal stress characteristics for different material variations and sizes of FGM. This basic study provides insight into the concept of FGM and lays the foundation of FGM optimization to control thermal stresses.

hp-Version discontinuous Galerkin methods for hyperbolic conservation laws

Abstract The development of hp -version discontinuous Galerkin methods for hyperbolic conservation laws is presented in this work. A priori error estimates are derived for a model class of linear hyperbolic conservation laws. These estimates are obtained using a new mesh-dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation. The results generalize and extend previous results on mesh-dependent norms to hp -version discontinuous Galerkin methods. A posteriori error estimates which provide bounds on the actual error are also developed in this work. Numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution.