Mesh refinement for shape optimization

The numerical solution of shape optimization problems is considered. The algorithm of successive optimization based on finite element techniques and design sensitivity analysis is applied. Mesh refinement is used to improve the quality of finite element analysis and the computed numerical solution. The norm of the variation of the Lagrange augmented functional with respect to boundary variation (residuals in necessary optimality conditions) is taken as an a posteriori error estimator for optimality conditions and the Zienkiewicz—Zhu error estimator is used to improve the quality of structural analysis. The examples presented show meaningful effects obtained by means of mesh refinement with a new error estimator.     

Velocity fields using NURBS with distortion control for structural shape optimization

The paper presents methods for the calculation of design velocity fields and mesh updating in the context of shape optimization. Velocity fields have a fundamental role in the integration of the main conceptual and software components in shape design optimization. Nonuniform rational B-splines are used to parameterize the domain boundary. A Newton/Raphson procedure is used to calculate the curve and surface internal parameters. A preconditioning iterative conjugate gradient method with low precision is used to improve the solution performance of the auxiliar problem in the calculation of the velocity fields. The velocity fields are also used to perturb the finite element mesh and element distortion measures are introduced. Finally, examples of two- and three-dimensional elastic problems are presented to illustrate the application of the algorithms.     

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 finite element alternative to infinite elements

In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method.     

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.     

Shape optimal design using B-splines

Shape optimal design of an elastic structure is formulated using a design element technique. It is shown that Bezier and B-spline curves, typical of the CAD philosophy, are well suited to the definition of design elements. Complex geometries can be described in a very compact way by a small set of design variables and a few design elements. Because of the B-splines flexibility, it is no longer necessary to piece design elements together in order to agree with the shape complexity, nor to restrict the shape variations. Moreover, the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh, are automatically taken into account in the new formulation. An analytical derivation of the sensitivity analysis will be established, giving rise to numerical efficiency. It will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables, so that simple mathematical programming algorithms can be applied to solve it. Some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper.     

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 Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

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.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Error estimation and mesh adaptation for Signorini–Coulomb problems using E-FEM

An error estimator for modeling contact and friction problems is presented in this paper. This estimator is obtained by solving two contact with friction problems: the first problem is formulated, as classically, in terms of displacement fields, and the second one is obtained using a stress field formulation. With this approach, it is necessary to develop a stress (equilibrium) finite element method such as that presented in previous studies. This estimator is similar to that discussed in [11]. The efficiency of the error estimator is tested by applying it to some examples. Due to the non-associativity of the friction problem, the present estimator is not strictly a majorant of the error. However, in the case of the examples studied here, the value of the estimator was approximately that of a given reference error. A refinement strategy was therefore developed. This strategy is very robust, even in the presence of stress singularities. With a sufficiently fine initial mesh, this method was found to be very efficient.     

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.     

Perimeter control in the bidirectional evolutionary optimization method

As an extension of the Evolutionary Structural Optimization [ESO] method, the bidirectional ESO [BESO] method has effectively addressed various topology optimization problems. It is observed in ESO/BESO applications that the optimal solution obtained strongly depends on the finite element mesh. This paper presents an improved BESO methodology incorporating a means of controlling the optimization process in order to satisfy a constraint on structural perimeter and thus to solve this size effect problem. Numerical tests are conducted on three examples at different levels of finite element discretization. Solutions convergent with respect to the grid refinement are obtained, especially when the perimeter bound is sufficiently low. Simpler topologies consisting of fewer and larger structural members and holes are generated when a more restricting constraint is imposed. Therefore, it is concluded that BESO combined with perimeter control is capable of suppressing mesh dependency and controlling the structural complexity for topology optimization problems.     

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.     

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

As has been shown in recent years, the approximate numerical differentiation of element stiffness matrices which is inherent in the semi-analytical method of finite element based design sensitivity analysis, may give rise to severely erroneous shape design sensitivities in static problems involving linearly elastic bending of beam, plate and shell structures.This paper demonstrates that the error problem also manifests itself in semi-analytical sensitivity analyses of eigenvalues of such structures and presents a method for complete elimination of the error problem. The method, which yields exact numerical sensitivities on the basis of simple first-order numerical differentiation, is computationally inexpensive and easy to implement as an integral part of the finite element analysis.The method is presented in terms of semi-analytical shape design sensitivity analysis of eigenvalues in the form of frequencies of free transverse vibrations of plates modelled by isoparametric Mindlin finite elements. Finally, the development is illustrated via two examples of occurrence of the error phenomenon when the traditional method is used and it is shown that the problem is completely eliminated by the application of the new method.     

Adaptive topology optimization

Topology optimization of continuum structures is often reduced to a material distribution problem. Up to now this optimization problem has been solved following a rigid scheme. A design space is parametrized by design patches, which are fixed during the optimization process and are identical to the finite element discretization. The structural layout is determined, whether or not there is material in the design patches. Since many design patches are necessary to describe approximately the structural layout, this procedure leads to a large number of optimization variables. Furthermore, due to a lack of clearness and smoothness, the results obtained can often only be used as a conceptual design idea.To overcome these shortcomings adaptive techniques, which decrease the number of optimization variables and generate smooth results, are introduced. First, the use of pure mesh refinement in topology optimization is discussed. Since this technique still leads to unsatisfactory results, a new method is proposed that adapts the effective design space of each design cycle to the present material distribution. Since the effective design space is approximated by cubic or Bézier splines, this procedure does not only decrease the number of design variables and lead to smooth results, but can be directly joined to conventional shape optimization. With examples for maximum stiffness problems of elastic structures the quality of the proposed techniques is demonstrated.     

A mixed variational formulation for shape optimization of solids with contact conditions

This paper is concerned with the development of a mixed variational formulation and computational procedure for the shape optimization problem of linear elastic solids in possible contact with a rigid foundation. The objective is to minimize the maximum value of the von Mises equivalent stress in a body (non-differentiable objective function), subject to a constraint on its volume and bound constraints on the design. For design purposes, the contact boundary is considered fixed.A finite element model that is appropriate for the mixed formulation is utilized in the discretization of the state and adjoint state equations. An elliptical mesh generator was used to generate the finite element mesh at each new design. The computational model is tested in several example problems.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

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.