Adaptive FE-procedures in shape optimization

In structural optimization the quality of the optimization result strongly depends on the reliability of the underlying structural analysis. This comprises the quality and range of the mechanical model, e.g. linear elastic or geometrically and materially nonlinear, as well as the accuracy of the numerical model, e.g. the discretization error of the FE-model. The latter aspect is addressed in the present contribution. In order to guarantee the quality of the numerical results the discretization error of the finite element solution is controlled and the finite element discretization is adaptively refined during the optimization process. Conventionally, so-called global error estimates are applied in structural optimization which estimate the error of the total strain energy. In the present paper local error estimates are introduced in shape optimization which allow to control directly the discretization error of local optimization criteria. In general, the adaptive refinement of the finite element discretization by remeshing affects the convergence of the optimization process if a gradient-based optimization algorithm is applied. In order to reduce this effect the sensitivity of the discretization error must also be restricted. Suitable refinement indicators are developed for globally and locally adaptive procedures. Finally, the potential of two techniques, which may improve the numerical efficiency of adaptive FE-procedures within the optimization process, is studied. The proposed methods and procedures are verified by 2-D shape optimization examples. Received June 3, 1999     

A simple error estimator for size and distortion of 2D isoparametric finite elements

The shape of elements in the finite element analysis may be the most important of many factors which induce a discretizing error. In particular, the efficiency of adaptive refinement analysis depends on the shape of the elements, and so estimating the quality of element shape is requisite during the adaptive analysis being performed. Unfortunately, most posterior error estimates can not evaluate the shape error of an element, so that some difficulties remain in the application of an adaptive analysis. For this purpose, an error estimator which can separately evaluate size error and distortion error from Zienkiewicz-Zhu's error estimator is presented for bilinear and quadratic isoparametric finite elements. As deduced from the results of numerical experiments, the suggeted estimator gives a reasonable evaluation of error due to element shape as well as discretizing error.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

A methodology for adaptive mesh refinement in optimum shape design problems

This work presents a methodology based on the use of adaptive mesh refinement (AMR) techniques in the context of shape optimization problems analyzed by the Finite Element Method (FEM). A suitable and very general technique for the parametrization of the optimization problem using B-splines to define the boundary is first presented. Then, mesh generation using the advancing front method, the error estimation and the mesh refinement criteria are dealt with in the context of a shape optimization problems. In particular, the sensitivities of the different ingredients ruling the problem (B-splines, finite element mesh, design behaviour, and error estimator) are studied in detail. The sensitivities of the finite element mesh coordinates and the error estimator allow their projection from one design to the next, giving an “a priori knowledge” of the error distribution on the new design. This allows to build up a finite element mesh for the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked out with some 2D examples.     

A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A residual type a posteriori error estimator is presented and analyzed for Weak Galerkin finite element methods for second order elliptic problems. The error estimator is proved to be efficient and reliable through two estimates, one from below and the other from above, in terms of an $H^1$ -equivalent norm for the exact error. Two numerical experiments are conducted to demonstrate the effectiveness of adaptive mesh refinement guided by this estimator.     

Adaptive multiple-levelh-refinement in automated finite element analyses

This paper discusses an automatic, adaptive finite element modeling system consisting of mesh generation, finite element analysis, and error estimation. The individual components interact with one another and efficiently reduce the finite element error to within an acceptable value and perform only a minimum number of finite element analyses.One of the necessary components in the automated system is a multiple-level local remeshing algorithm. Givenh-refinement information provided by an a posteriori error estimator, and adjacency information available in the mesh data structures, the local remeshing algorithm grades the refinement toward areas requesting refinement. It is shown that the optimal asymptotic convergence rate is achieved, demonstrating the effectiveness of the intelligent multiple-level localh-refinement.     

A posteriori error estimation in sensitivity analysis

We present a posteriori error estimators suitable for automatic mesh refinement in the numerical evaluation of sensitivity by means of the finite element method. Both diffusion (Poisson-type) and elasticity problems are considered, and the equivalence between the true error and the proposed error estimator is proved. Application to shape sensitivity is briefly addressed.     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

A posteriori optimization of parameters in stabilized methods for convection–diffusion problems – Part I

Stabilized finite element methods for convection-dominated problems require the choice of appropriate stabilization parameters. From numerical analysis, often only their asymptotic values are known. This paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional. Exemplarily, this framework is applied to the SUPG finite element method and the minimization of a residual-based error estimator, an error indicator, and a functional including the crosswind derivative of the computed solution. Benefits of the basic approach are demonstrated by means of numerical results.     

An a posteriori error estimator for hp-adaptive continuous Galerkin methods for photonic crystal applications

In this paper we propose and analyse an error estimator suitable for $hp$ -adaptive continuous finite element methods for computing the band structure and the isolated modes of 2D photonic crystal (PC) applications. The error estimator that we propose is based on the residual of the discrete problem and we show that it leads to very fast convergence in all considered examples when used with $hp$ -adaptive refinement techniques. In order to show the flexibility and robustness of the error estimator we present an extensive collection of numerical experiments inspired by real applications. In particular we are going to consider PCs with point defects, PCs with line defects, bended waveguides and semi-infinite PCs.     

Feasibility refinement in sequential quadratic programming using parametric sensitivity analysis

In this paper we present results that extend the sequential quadratic programming (SQP) algorithm with an additional feasibility refinement based on parametric sensitivity derivatives. The refinement is applicable without restriction on the problem dimensions in sparse SQP solvers. Parametric sensitivity analysis is a tool for post optimality analysis of the solution of a nonlinear optimization problem. For the refinement approach we apply this technique on the quadratic subproblems in order to improve the overall algorithm. The sensitivity derivatives required for this approach can be computed without noticeable computational effort as the system of linear equations to be solved coincides with the system already solved for the search direction computation. For similar algorithms in the context of post optimality analysis a linear rate of convergence has been proven and therefore an extrapolation method is applied to speed up the process. The presented algorithm has been integrated into the nonlinear program (NLP) solver WORHP and we perform a numerical study to evaluate different termination criteria for the proposed algorithm. Furthermore, numerical results on the CUTEst test set are shown.     

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.     

Error control and mesh optimization for high-order finite element approximation of incompressible viscous flow

We discuss the use of a posteriori error estimates for high-order finite element methods during simulation of the flow of incompressible viscous fluids. The correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency. We show how to use the error estimator for mesh optimization which improves computational efficiency for both steady-state and unsteady flows. The method is applied to two-dimensional problems with known analytical solutions (Jeffrey-Hamel flow) and more complex flows around a body, both in a channel and in an open domain.     

An error analysis approach for laminated composite plate finite element models

Errors in laminated composite plate finite element models occur at both the individual element level and at the discretization level. This paper shows that parasitic shear causes individual element errors and that its sources must be eliminated if numerically and physically correct results are to be provided by the finite element analysis. In addition, discretization errors occur when the behavior of the continuum is represented by a finite number of degrees of freedom. A procedure to estimate discretization errors in laminated composite plate finite element models and guide refinement, in order to achieve an acceptable level of accuracy, is developed. The error estimator built is based on the energy norm of the error in stress resultants.     

A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which, though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An “equivalent estimator” for the H¹ finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.     

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

A space–time adaptation scheme for unsteady shallow water problems

We provide a space–time adaptation procedure for the approximation of the Shallow Water Equations (SWE). This approach relies on a recovery based estimator for the global discretization error, where the space and time contributions are kept separate. In particular we propose an ad hoc procedure for the recovery of the time derivative of the numerical solution and then we employ this reconstruction to define the error estimator in time. Concerning the space adaptation, we move from an anisotropic error estimator able to automatically identify the density, the shape and the orientation of the elements of the computational mesh. The proposed global error estimator turns out to share the good properties of each recovery based error estimator. The whole adaptive procedure is then combined with a suitable stabilized finite element SW solver. Finally the reliability of the coupled solution–adaptation procedure is successfully assessed on two unsteady test cases of interest for hydraulics applications.     

On the verification of model reduction methods based on the proper generalized decomposition

In this work, we introduce a consistent error estimator for numerical simulations performed by means of the proper generalized decomposition (PGD) approximation. This estimator, which is based on the constitutive relation error, enables to capture all error sources (i.e. those coming from space and time numerical discretizations, from the truncation of the PGD decomposition, etc.) and leads to guaranteed bounds on the exact error. The specificity of the associated method is a double approach, i.e. a kinematic approach and a unusual static approach, for solving the parameterized problem by means of PGD. This last approach makes straightforward the computation of a statically admissible solution, which is necessary for robust error estimation. An attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes; it thus represents a practical and relevant tool for driving algorithms carried out in PGD, being possibly used as a stopping criterion or as an adaptation indicator. Numerical experiments on transient thermal problems illustrate the performances of the proposed method for global error estimation.     

The Superconvergent Cluster Recovery Method

A new gradient recovery technique SCR (Superconvergent Cluster Recovery) is proposed and analyzed for finite element methods. A linear polynomial approximation is obtained by a least-squares fitting to the finite element solution at certain sample points, which in turn gives the recovered gradient at recovering points. Compared with similar techniques such as SPR and PPR, our approach is cheaper and efficient, while having same or even better accuracy. In additional, it can be used as an a posteriori error estimator, which is relatively simple to implement, cheap in terms of storage and computational cost for adaptive algorithms. We present some numerical examples illustrating the effectiveness of our recovery procedure.     

SUT-DAM: An integrated software environment for multi-disciplinary geotechnical engineering

As computer simulation increasingly supports engineering design, the requirement for a computer software environment providing an integration platform for computational engineering software increases. A key component of an integrated environment is the use of computational engineering to assist and support solutions for complex design. In the present paper, an integrated software environment is demonstrated for multi-disciplinary computational modeling of structural and geotechnical problems. The SUT-DAM is designed in both popularity and functionality with the development of user-friendly pre- and post-processing software. Pre-processing software is used to create the model, generate an appropriate finite element grid, apply the appropriate boundary conditions, and view the total model. Post-processing provides visualization of the computed results. In SUT-DAM, a numerical model is developed based on a Lagrangian finite element formulation for large deformation dynamic analysis of saturated and unsaturated soils. An adaptive FEM strategy is used into the large displacement finite element formulation by employing an error estimator, adaptive mesh refinement, and data transfer operator. This consists in defining new appropriate finite element mesh within the updated, deformed geometry and interpolating (mapping) the pertinent variables from one mesh to another in order to continue the analysis. The SUT-DAM supports different yield criteria, including classical and advanced constitutive models, such as the Pastor–Zienkiewicz and cap plasticity models. The paper presents details of the environment and includes several examples of the integration of application software.