Adaptive mesh method for topology optimization of fluid flow

In order to solve the topology optimization problems of fluid flow and obtain higher resolution of the interface with a minimum of additional expense, an automatic local adaptive mesh refinement method is proposed. The optimization problem is solved by a simple but robust optimality criteria (OC) algorithm. A material distribution information based adaptive mesh method is adopted during the optimization process. The optimization procedure is provided and verified with several benchmark examples.     

Comparison of refinement criteria for structured adaptive mesh refinement

We have investigated and analyzed the grid convergence issues for an adaptive mesh refinement (AMR) code. We have found that the numerical results for the AMR grid may have a larger error than those for the unrefined uniform grid. After a detailed analysis, we have found that the numerical solution at the coarse-fine interface between different levels of the grid converges only in the first-order accuracy. Therefore, the error near the coarse-fine interface can quickly dominate the error in the other regions if the coarse-fine interface is active and not covered by the fine grid. We propose, implement, and compare several refinement criteria. Some of them can catch the large-error region near the coarse-fine interface and refine them with the fine grid.     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

Optimal shape design for fluid flow using topological perturbation technique

This paper is concerned with an optimal shape design problem in fluid mechanics. The fluid flow is governed by the Stokes equations. The theoretical analysis and the numerical simulation are discussed in two and three-dimensional cases. The proposed approach is based on a sensitivity analysis of a design function with respect to the insertion of a small obstacle in the fluid flow domain. An asymptotic expansion is derived for a large class of cost functions using small topological perturbation technique. A fast and accurate numerical algorithm is proposed. The efficiency of the method is illustrated by some numerical examples.     

The Limits of Porous Materials in the Topology Optimization of Stokes Flows

We consider a problem concerning the distribution of a solid material in a given bounded control volume with the goal to minimize the potential power of the Stokes flow with given velocities at the boundary through the material-free part of the domain.We also study the relaxed problem of the optimal distribution of the porous material with a spatially varying Darcy permeability tensor, where the governing equations are known as the Darcy–Stokes, or Brinkman, equations. We show that the introduction of the requirement of zero power dissipation due to the flow through the porous material into the relaxed problem results in it becoming a well-posed mathematical problem, which admits optimal solutions that have extreme permeability properties (i.e., assume only zero or infinite permeability); thus, they are also optimal in the original (non-relaxed) problem. Two numerical techniques are presented for the solution of the constrained problem. One is based on a sequence of optimal Brinkman flows with increasing viscosities, from the mathematical point of view nothing but the exterior penalty approach applied to the problem. Another technique is more special, and is based on the “sizing” approximation of the problem using a mix of two different porous materials with high and low permeabilities, respectively. This paper thus complements the study of Borrvall and Petersson (Internat. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003), where only sizing optimization problems are treated.     

Composite penalty method of a low order anisotropic nonconforming finite element for the Stokes problem

Composite penalty method of a low order anisotropic nonconforming quadrilateral finite element for the Stokes problem is presented. This method with a large penalty parameter can achieve the same accuracy as the stand method with a small penalty parameter and the convergence rate of this method is two times as that of the standard method under the condition of the same order penalty parameter. The superconvergence for velocity is established as well. The results of this paper are also valid to the most of the known nonconforming finite element methods.     

Shape optimal design of arch dams using an adaptive neuro-fuzzy inference system and improved particle swarm optimization

An efficient methodology is proposed to find the optimal shape of arch dams including fluid–structure interaction subject to earthquake ground motion. In order to reduce the computational cost of optimization process, an adaptive neuro-fuzzy inference system (ANFIS) is built to predict the dam effective response instead of directly evaluating it by a time-consuming finite element analysis (FEA). The presented ANFIS is compared with a widespread neural network termed back propagation neural network (BPNN) and it appears a better performance generality for estimating the dam response. The optimization task is implemented using an improved version of particle swarm optimization (PSO) named here as IPSO. In order to assess the effectiveness of the proposed methodology, the optimization of a real world arch dam is performed via both IPSO–ANFIS and PSO–BPNN approaches. The numerical results demonstrate the computational advantages of the proposed IPSO–ANFIS for optimal design of arch dams when compared with the PSO–BPNN approach.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

Flow simulation and shape optimization for aircraft design

Within the framework of the German aerospace research program, the CFD project MEGADESIGN was initiated. The main goal of the project is the development of efficient numerical methods for shape design and optimization. In order to meet the requirements of industrial implementations a co-operative effort has been set up which involves the German aircraft industry, the DLR, several universities and some small enterprises specialized in numerical optimization. This paper outlines the planned activities within MEGADESIGN, the status at the beginning of the project and it presents some early results achieved in the project.     

Unified finite element discretizations of coupled Darcy–Stokes flow

In this article, we discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy's law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods proposed here utilize the same finite element spaces on the entire domain. In particular, we show how the coupled problem can be solved by using standard Stokes elements like the MINI element or the Taylor–Hood element in the entire domain. Furthermore, for all the methods the handling of the interface conditions are straightforward. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An adaptive mesh method for 1D hyperbolic conservation laws

An adaptive method is developed for solving one-dimensional systems of hyperbolic conservation laws, which combines the rezoning approach with the finite volume weighted essentially non-oscillatory (WENO) scheme. An a posteriori error estimate, used to equidistribute the mesh, is obtained from the differences between respective numerical solutions of 5th-order WENO (WENO5) and 3rd-order ENO (ENO3) schemes. The number of grids can be adaptively readjusted based on the solution structure. For higher efficiency, mesh readjustment is performed every few time steps rather than every time step. In addition, a high order conservative interpolation is used to compute the physical solutions on the new mesh from old mesh based on the finite volume ENO reconstruction. Extensive examples suggest that this adaptive method exhibits more accurate resolution of discontinuities for a similar level of computational time comparing with that on a uniform mesh.     

A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow

Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.      

An adaptive boundary element method for the exterior Stokes problem in three dimensions

^** Email: vjervin{at}clemson.edu^*** Email: norbert.heuer{at}brunel.ac.uk We present an adaptive refinement strategy for the h-versionof the boundary element method with weakly singular operatorson surfaces. The model problem deals with the exterior Stokesproblem, and thus considers vector functions. Our error indicatorsare computed by local projections onto 1D subspaces definedby mesh refinement. These indicators measure the error separatelyfor the vector components and allow for component-independentadaption. Assuming a saturation condition, the indicators giverise to an efficient and reliable error estimator. Also we describehow to deal with meshes containing quadrilaterals which arenot shape regular. The theoretical results are underlined bynumerical experiments. To justify the saturation assumption,in the Appendix we prove optimal lower a priori error estimatesfor edge singularities on uniform and graded meshes.     

Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

Existence analysis of an optimal shape design problem with non coercive state equation

In this paper, a shape optimization problem, modelling a welding process, governed by a second order non coercive PDE is considered. The well posedness of the shape optimal design problem is showed using the degree of Leray–Schauder. Then the existence of an optimal solution is proved.     

Stokes型积分-微分方程的Crouzeix-Raviart型非协调三角形各向异性有限元方法

在半离散格式下.研究了Stokes型积分一微分方程的Crouzeix-Raviart型非协调三角形各向异性有限元方法,在不需要传统Ritz-Volterra投影下,通过辅助空间等新的技巧得到了与传统有限元方法相同的误差估计.     

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X_H and X_h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ? H, respectively, and a finite element space M_h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H⁵. If we choose H = O(h^2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.