HIGH-PERFORMANCE PCG SOLVERS FOR FEM STRUCTURAL ANALYSIS

The preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems. Such systems are generated by the finite element discretization in structural analysis but users of finite elements in this context generally still rely on direct methods. It is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods.     

Finite element and conjugate gradient methods for a nonlinear optimal heat transfer control problem

The boundary control problem of optimal heating of an infinitely long slab with tempertue-dependent thermal conductivity, subjected to a convection and radiation boundary condition, is analysed by numerical methods. In order to reformulate the optimal control problem of distributed parameter systems as a mathematical programming problem of finite dimension, a space, co-ordinate is discretized by use of the finite element method, while the Runge–Kutta method is utilized for time integrations. Finally, the performance index of the optimal control problem is minimized by the conjugate gradient method of optimization.     

A gradient‐only line search method for the conjugate gradient method applied to constrained optimization problems with severe noise in the objective function

A new implementation of the conjugate gradient method is presented that economically overcomes the problem of severe numerical noise superimposed on an otherwise smooth underlying objective function of a constrained optimization problem. This is done by the use of a novel gradient‐only line search technique, which requires only two gradient vector evaluations per search direction and no explicit function evaluations. The use of this line search technique is not restricted to the conjugate gradient method but may be applied to any line search descent method. This method, in which the gradients may be computed by central finite differences with relatively large perturbations, allows for the effective smoothing out of any numerical noise present in the objective function. This new implementation of the conjugate gradient method, referred to as the ETOPC algorithm, is tested using a large number of well‐known test problems. For initial tests with no noise introduced in the objective functions, and with high accuracy requirements set, it is found that the proposed new conjugate gradient implementation is as robust and reliable as traditional first‐order penalty function methods. With the introduction of severe relative random noise in the objective function, the results are surprisingly good, with accuracies obtained that are more than sufficient compared to that required for engineering design optimization problems with similar noise. Copyright © 2004 John Wiley & Sons, Ltd.     

Comparison between element,edge and compressed storage schemes for iterative solutions in finite element analyses

This paper is concerned with optimization techniques for the iterative solution of sparse linear systems arising from finite element discretization of partial differential equations. Three different data structures are used to store the coefficient matrices: the usual element‐based data structure, the compressed storage row format and the edge‐based approach. A comparison between these storage schemes is performed, quantifying for most common linear elements the number of floating points operations, indirect addressing and memory requirements necessary to perform matrix–vector products. The overall performance of the preconditioned conjugate gradient method is measured for different situations involving 2D and 3D diffusion and elasticity problems, highlighting the pros and cons of each storage scheme. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element solution of a tidal current problem in the seto inland sea by using the ICCG method

When the finite element method is applied to the analysis of tidal currents in an inland sea with many islands, a system of linear equations with large band and sparse coefficient matrix is solved at each time step, and therefore the finite element methods usually suffer a severe economic disadvantage for practical calculations. The method used in this paper for solving a system of linear equations with large band and sparse coefficient matrix is the incomplete Cholesky conjugate gradient (ICCG) method: The ICCG method was compared with other methods such as the Gaussian elimination method, the Gauss–Seidel method and the conjugate gradient method. This method showed significant improvement in computation time and it can overcome the disadvantage that the efficiency to solve the matrix equations which appear in the finite element analysis of tidal currents usually diminishes as the bandwidth grows. The simulation results of tidal currents in the Seto Inland Sea of Japan were compared with field data and good agreements were obtained.     

A comparison of Lanczos and optimization methods in the partial solution of sparse symmetric eigenproblems

In the present paper, we analyse the computational performance of the Lanczos method and a recent optimization technique for the calculation of the p (p ≤ 40) leftmost eigenpairs of generalized symmetric eigenproblems arising from the finite element integration of elliptic PDEs. The accelerated conjugate gradient method is used to minimize successive Rayleigh quotients defined in deflated subspaces of decreasing size. The pointwise Lanczos scheme is employed in combination with both the Cholesky factorization of the stiffness matrix and the preconditioned conjugate gradient method for evaluating the recursive Lanczos vectors. The three algorithms are applied to five sample problems of varying size up to almost 5000. The numerical results show that the Lanczos approach with Cholesky triangularization is generally faster (up to a factor of 5) for small to moderately large matrices, while the optimization method is superior for large problems in terms of both storage requirement and CPU time. In the large case, the Lanczos–Cholesky scheme may be very expensive to run even on modern quite powerful computers.     

Cost reduction techniques for the design of non‐linear flapping wing structures

This work details a computational framework for gradient‐based optimization of a non‐linear flapping wing structure with a large number of design variables, where analytical sensitivities of the unsteady finite element system are computed using the adjoint method. Two techniques are used to reduce the large computational cost of this structural design process. The first projects the finite element system onto a reduced basis of POD modes. The second uses a monolithic time formulation with spectral elements, and can be used to compute only the desired time‐periodic response. Results are given in terms of the trade‐off between accuracy and computational efficiency of these methods for both system response and adjoint computations, for a variety of mesh/time step refinements, degrees of non‐linearity (i.e. weakly or strongly non‐linear), and harmonic content. The work concludes with the structural design of a flapping wing: the elastic deformation at the wingtip is minimized through the flapping stroke by varying the thickness of each finite element. Significant improvements in computational cost are obtained at little expense to the accuracy of the results obtained via design optimization. Published in 2011 by John Wiley & Sons, Ltd.     

基于PVM的网络并行子结构共轭梯度法

网络并行环境是近年来国际上并行环境的一个重要方向,ＰＶＭ是当前最流行的支持异构或同构型网络并行计算的软件平台之一。本文采用子结构共轭梯度法研究了基于ＰＶＭ的网络并行有限元,该方法将有限元网格划分为ｎ个子结构,再将ｎ个子结构的数据分送给网上ｎ台可用微机,ｎ台微机并行形成和组集ｎ个子结构的劲度矩阵和荷载列阵,然后采用预条件共轭梯度法并行求解结点位移,最后ｎ台微机并行对ｎ个子结构进行应变和应力分析。该方法不需形成结构的总体劲度矩阵和荷载列阵,可同时迭代求出所有结点位移,且比一般的迭代法收敛要快。算例表明此种并行子结构共轭梯度法在网络上能获得较高的并行加速比。     

Topology optimization of finite strain viscoplastic systems under transient loads

A transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.     

Elastic-plastic analysis of crack in adhesive joint by combination of boundary element and finite element methods

This paper concerns with a combination method of the boundary element method and the finite element method for elastic-plastic analyses. The combination method proposed here is based on the substructure method using the conjugate gradient method. This combination method has the advantage of saving CPU time and memory storage size over the finite element method. The combination method is applied to a J-integral analysis of a crack in an adhesive joint. The effect of bond thickness on the J-integral is discussed.     

Preconditioned conjugate- and secant-Newton methods for non-linear problems

The preconditioned conjugate gradient (CG) method is becoming accepted as a powerful tool for solving the linear systems of equations resulting from the application of the finite element method. Applications of the non-linear algorithm are mainly confined to the diagonally scaled CG. In this study the coupling of preconditioning techniques with non-linear versions of the conjugate gradient and quasi-Newton methods creates a set of conjugate- and secant-Newton methods for the solution of non-linear problems. The preconditioning matrices used to improve the ellipticity of the problem and to reduce the computer storage requirements are obtained by the application of the partial preconditioning and the partial elimination techniques. Both techniques use a drop-off parameter ψ to control the computer storage demands of the method, making it more versatile for any computer hardware environment. Consideration is given to the development of a highly effective stability test for the line search minimization routine, which computes accurate values without much effort. This results in a beneficiary effect not only on the convergence properties of the methods but on their efficiency as well.     

Symmetric FE–BE coupling with iterations on the interface

The paper presents an algorithm for the solution of problems that are discretized partly by the finite element method and partly by the boundary element method. The algorithm is based on the conjugate gradient method with preconditioning by an auxiliary conjugate projector that reduces the iterations to the interface. A numerical example is presented to illustrate the performance of the algorithm. The method may prove useful also in parallel environment.     

A theoretical and computational model of eddy-current probesincorporating volume integral and conjugate gradient methods

A general three-dimensional computational model of ferrite-core eddy-current probes has been developed for research and design studies in nondestructive evaluation. The model is based on a volume integral approach for finding the magnetization of the ferrite core excited by an AC-current-carrying coil in the presence of a conducting workpiece. Using the moment method, the integral equation is approximated by a matrix equation and solved using conjugate gradient techniques. Illustrative results are presented showing the impedance characteristics and field distributions for practical eddy-current probe configurations     

A general methodology for structural shape optimization problems using automatic adaptive remeshing

The proposed methodology is based on the use of the adaptive mesh refinement (AMR ) techniques in the context of 2D shape optimization problems analysed by the finite element method. 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 frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines. finite element mesh, structural behaviour and error estimator) is studied in detail. The sensitivities of the finite element mesh and error estimator permit their projection from one design to the next one leading to an a priori knowledge of the finite element error distribution on the new design without the necessity of any additional structural analysis. With this information the mesh refinement criterion permits one to build up a finite element mesh on the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked by means of several examples.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive solution strategy for solving large systems of p-type finite element equations

A solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p-type finite element method using a hierarchical basis function set. The algorithm dynamically branches to either direct or iterative solution methods. In. the iterative solution branch, the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient (PCG) method. The convergence rate of the PCG algorithm is monitored to improve the heuristics used in the choice of the preconditioner. The robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations. This strategy has been implemented in a p-version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems.     

基于密度分布曲线法的复合材料变刚度铺层优化

基于梯度的优化方法对复合材料层合板进行了变刚度铺层优化设计。在优化过程中需确定铺层中各单元的密度以及角度。为了使优化结果具有可制造性,优化结果需满足制造工艺约束并且铺层角度需从预定角度中选取。为了避免在优化问题中引入过多的约束并减少设计变量的数目,提出密度分布曲线法(DDCM)对层合板中各单元的密度进行参数化。根据各单元的密度以及角度设计变量并基于Bi-value Coding Parameterization(BCP)方法中的插值公式确定各单元的弹性矩阵。优化过程中以结构柔顺度作为优化目标,结构体积作为约束,优化算法采用凸规划对偶算法。对碳纤维复合材料的算例结果表明:采用DDCM可得到较理想的优化结果,并且收敛速率较快。     

BEM and FEM combination parallel analysis using conjugate gradient and condensation

Boundary element and finite element combination analysis on parallel schemes are improved in this paper. The conjugate gradient method (CG method) is introduced for renewal of unknowns on the combination boundary in place of the Schwarz method previously used, which makes it possible to determine a parameter required in the renewal iteration automatically. Further, the condense method is employed for higher efficiency of solution by reducing the number of degree of freedoms in both equations for the finite element and boundary element domains. Comparison of the present algorithm with the previous one in some numerical examples shows marked improvement in computational efficiency.     

Hybrid FE–IDQ–CG method for dynamic parameters estimation of multilayered half-space

As a first endeavor, a hybrid finite element (FE)–incremental differential quadrature (IDQ) method together with the discrepancy principle and the conjugate gradient method (CGM) is used to develop an inverse algorithm for the parameters estimation of the axisymmetric multilayered half-spaces. The approach is based on the measurement of the dynamic transverse displacement at some boundary points of the half-space to estimate the unknown parameters of its layers. Using the accuracy and unconditional stability of the hybrid FE–IDQ method, the direct problem is solved to get the dynamic transverse displacements. After adding some random errors to the obtained results, they are considered as the measured responses by sensors. Then, the conjugate gradient method as a general and robustness optimization technique is employed to minimize the error between the measured and calculated dynamic surface responses at sensor locations. The sensitivity analysis of the displacement field is performed using a semi-analytical method. The applicability and correctness of the proposed hybrid algorithm is demonstrated through different examples by considering the influence of the layers arrangement, the measurement errors and sensor numbers.     

On a new edge‐based gradient recovery technique

A posteriori error estimation methods for mesh adaptation often require an accurate computation of the gradient of a Lagrange finite element solution. The precision of the error estimation is directly related to the accuracy of the recovered gradient. We therefore present in this communication a simple method for the evaluation of the gradient of a linear Lagrange finite element solution, and we show that it has significant advantages over existing methods of the same order. The proposed method requires the solution of a global linear system that can be solved by a preconditioned conjugate gradient method. An interesting feature is that it does not require any special treatment for boundary nodes contrarily to classical local patch recovery methods. Copyright © 2012 John Wiley & Sons, Ltd.