Advances in Adaptive Computational Methods in Mechanics

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.

Boundary Integral Equations and A Posteriori Error Estimates

A review of some a posteriori error estimates for adaptive finite element methods

Smart computational fracture of materials and structures

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017

In this work, we combine an hp–adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction.Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp–adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

In the paper we investigate first the problem of a posteriori error estimate for the Navier-Stokes equations to model the flow of incompressible fluid in a 2D domain. We discuss the role of the constant in the a posteriori estimate. Then we apply the a posteriori error estimate with the constant found numerically, to the adaptive mesh refinement. We present numerical results in case of an incompressible flow problem in a domain with corners that cause singularities in the solution. We also show an alternative approach based on the asymptotic behavior of the solution near nonconvex corners.

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard dual-weighted-residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

An adaptive finite element procedure is presented for improving the quality of solutions to convection‐dominated problems in geodynamics. The method adapts the mesh automatically around regions of high solution gradient, yielding enhanced resolution of the associated flow features. The approach requires the coupling of an automatic mesh generator, a finite element flow solver, and an error estimator. In this study, the procedure is implemented in conjunction with the well‐known geodynamical finite element code ConMan. An unstructured quadrilateral mesh generator is utilized, with mesh adaptation accomplished through regeneration. This regeneration employs information provided by an interpolation‐based local error estimator, obtained from the computed solution on an existing mesh. The technique is validated by solving thermal and thermochemical problems with well‐established benchmark solutions. In a purely thermal context, results illustrate that the method is highly successful, improving solution accuracy while increasing computational efficiency. For thermochemical simulations the same conclusions can be drawn. However, results also demonstrate that the grid‐based methods employed for simulating the compositional field are not competitive with the other methods (tracer particle and marker chain) currently employed in this field, even at the higher spatial resolutions allowed by the adaptive grid strategies.

The purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so-called pointwise tracking optimal control problem. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state, thus leading to an adjoint problem with Dirac measures on the right hand side; control constraints are also considered. The proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in Muckenhoupt weighted Sobolev spaces for the adjoint state. We present an analysis that is valid for two and three-dimensional domains. We conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator.

We present a parallel goal‐oriented adaptive finite element algorithm that can be used to rapidly compute highly accurate solutions for 2.5D controlled‐source electromagnetic (CSEM) and 2D magnetotelluric (MT) modeling problems. The use of an unstructured triangular modeling grid allows for efficient use of mesh nodes for discretizing arbitrarily complex domains. A goal‐oriented error estimator based on the dual residual weighting method computed through hierarchical bases provides robust error estimation that is used to guide an iterative adaptive mesh refinement process. Our formulation of the error estimator considers the relative error in the strike aligned fields and their spatial gradients, and therefore results in a more efficient use of mesh nodes than previous error estimators based on absolute field errors. This algorithm has been parallelized over frequencies, transmitters, receivers and wave‐numbers, enabling it to achieve accurate solutions in run‐times of seconds to tens of seconds for realistic models and data parameters when run on cluster computers of up to a thousand processors. Application of this new algorithm to a complex model that includes strong seafloor topography variations and multiple thin stacked reservoirs demonstrates the performance and scalability on a large cluster computer.

We compute solutions of solutal phase-field models for dendritic growth of an isothermal binary alloy using anisotropic mesh refinement techniques. The adaptive strategy is based on anisotropic a posteriori estimators using a superconvergent recovery technique in the form of the Zienkiewicz-Zhu error estimator. The phase-field model contains an anisotropic strongly nonlinear second order operator modelling the dendritic branches, this strong nonlinearity is included in the a posteriori error estimators by using a monotonicity result. The monotonicity holds for phase-field anisotropy below a certain threshold value beyond which there are no known well-posedness results. We present computational results for both regimes showing the performance of the proposed method.

Mesh adaptation is essential for accurate computational fluid dynamics (CFD) simulations, especially when computational resources are limited.In simulating the advection-diffusion equation using the finite volume method (FVM), mesh adaptation is often necessary to accurately capture sharp gradients and complex flow features.Although the metric-based mesh adaptation used in this study is highly effective with true errors, these are typically unavailable, requiring the use of error estimators.However, adjoint-based error estimators, though effective, are computationally expensive as they require solving the larger adjoint system.To address this, an inexpensive error estimator based on the gradient of the numerical solution and the embedded method is proposed.Specifically, the central difference and upwind difference schemes are utilized for error estimation within this framework.Since errors are predominantly influenced by sharp gradients in the scalar boundary layer, a target functional based on the product of the embedded error and the solution gradient to compute the error estimator is proposed.Consequently, these proposed error estimators generate meshes closely resembling those produced using true error values, effectively resolving the scalar boundary layer.As a result, the L 2 and L norms of the error typically decrease with successive adaptation cycles.Furthermore, the L 1 , L 2 , and L norms of error per element also reduce with successive adaptation cycles.In conclusion, the proposed error estimator facilitates mesh adaptation without needing true errors or computationally expensive adjoint-based error estimators, thus ensuring accurate numerical solutions, especially in scenarios with sharp boundary layers.

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