Approximate reanalysis in topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures is investigated. The nested optimization problem is reformulated to accommodate the use of an approximate displacement vector and the design sensitivities are derived accordingly. It is shown that relatively rough approximations are acceptable since the errors are taken into account in the sensitivity analysis. The implementation is tested on several small and medium scale problems, including 2‐D and 3‐D minimum compliance problems and 2‐D compliant force inverter problems. Accurate results are obtained and the savings in computation time are promising. Copyright © 2008 John Wiley & Sons, Ltd.     

基于人工材料的结构拓扑渐进优化设计

首先,提出了一种在结构边界和孔洞周围附加人工材料的思路。在此基础上,结合ESO方法和应力灵敏度,建立了结构有限单元增、删的准则, 给出了一种新的拓扑优化算法。算例表明该方法能采用固定有限元网格中不同的初始优化结构就可获得优化拓扑。由于其概念上的简洁性和应用上的有效性,该方法具有一定的工程应用价值。     

A finite element approach combining a reduced‐order system,Padé approximants,and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

In this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 John Wiley & Sons, Ltd.     

梁杆结构几何非线性有限元的数值实现方法

梁杆结构几何非线性有限元方法主要包括两个部分, 建立虚功方程和实现数值求解. 该文运用对比方法, 分析了采用UL型增量理论的梁杆结构几何非线性有限元法求解过程与连续体求解过程的主要不同点, 特别是论述了确定加载步末的内力状态的重要性和方法.     

Sizing design sensitivity analysis of non-linear structural systems. Part II: Numerical method

A numerical method is presented to carry out sizing design sensitivity calculations outside established finite element analysis codes, using postprocessing data only. Geometric as well as material non-linearities are treated. To demonstrate the accuracy of the proposed method, numerical results are presented for structural systems with linear elastic material, large displacements, large rotations and small strains. A distributed parameter approach to structural design sensitivity analysis is used to retain the continuum elasticity formulation throughout the derivation of design sensitivity results. Using this approach and an adjoint variable method, design sensitivity computations are carried out. For structural performance functionals stress and displacement are considered. It is shown that computations can be performed with the same computational effort as for sizing design sensitivity analysis of linear structural systems. Accurate design sensitivity results are obtained for both linear and non-linear structural systems without the uncertainty of numerical accuracy and high cost associated with the selection of finite difference perturbations. Also, the method does not require differentiation of element stiffness and mass matrices in conventional finite element models.     

非线性动力方程的精细时空有限元方法

根据Hamilton变作用定律构造了时空有限元矩阵;并根据传递矩阵原理,利用时间的一维性将时空有限元矩阵变换为时间方向的传递矩阵,将初值问题转化为一般矩阵相乘问题以方便求解。为了保证计算的稳定性,参考了精细积分的思想提出精细时空有限元方法,并给出线性问题在时间级数荷载作用下的计算式。数值分析结果证明该方法在线性问题分析上非常准确并可以推广到非线性动力方程的求解;只需将非线性解看作初始解和增量解的叠加,通过精细时空有限元线性求解方法计算增量解,逐步修正后即可得到非线性解。结果表明该方法是一个有效的求解非线性动力方程的方法。     

A finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Enriching the finite element method with meshfree particles in structural mechanics

The automatic generation of meshes for the finite element (FE) method can be an expensive computational burden, especially in structural problems with localized stress peaks. The use of meshless methods can address such an issue, as these techniques do not require the existence of an underlying connection among the particles selected in a general domain. This study advances a numerical strategy that blends the FE method with the meshless local Petrov–Galerkin technique in structural mechanics, with the aim at exploiting the most attractive features of each procedure. The idea relies on the use of FEs to compute a background solution that is locally improved by enriching the approximation space with the basis functions associated to a few meshless points, thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid. Adding the meshless particles only where needed avoids the cost of mesh refining, or even of remeshing, without the prohibitive computational cost of a thoroughly meshfree approach. In the present implementation, an efficient integration strategy for the computation of the coefficients taking into account the mutual FE–meshless local Petrov–Galerkin interactions is introduced. Moreover, essential boundary conditions are enforced separately on both FEs and meshless particles, thus allowing for an overall accuracy improvement also when the enriched region is close to the domain boundary. Numerical examples in structural problems show that the proposed approach can significantly improve the solution accuracy at a local level, with no remeshing effort, and at a low computational cost. Copyright © 2016 John Wiley & Sons, Ltd.     

A monolithic strategy for fluid–structure interaction problems

In this work, we present a new monolithic strategy for solving fluid–structure interaction problems involving incompressible fluids, within the context of the finite element method. This strategy, similar to the continuum dynamics, conserves certain properties, and thus provides a rational basis for the design of the time‐stepping strategy; detailed proofs of the conservation of these properties are provided. The proposed algorithm works with displacement and velocity variables for the structure and fluid, respectively, and introduces no new variables to enforce velocity or traction continuity. Any existing structural dynamics algorithm can be used without change in the proposed method. Use of the exact tangent stiffness matrix ensures that the algorithm converges quadratically within each time step. An analytical solution is presented for one of the benchmark problems used in the literature, namely, the piston problem. A number of benchmark problems including problems involving free surfaces such as sloshing and the breaking dam problem are used to demonstrate the good performance of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

3-D magnetic field computations using finite-element approach with localized functional

An extended application of a finite-element approach with localized functional to a three-dimensional magnetic field problem is described in this paper. The field region is modeled by a set of partial differential equations in terms of scalar potentials. The variational approach is used to obtain the system matrix. The localized functional is derived, which consists of the domain integral of the finite element region only and the boundary integral of the interfacial boundary between the finite and infinite element regions. The proposed approach is applied to a sample problem. The result has been compared with the standard finite element method and an analytic solution. The numerical solutions obtained by the proposed approach are in good agreement with the analytic solutions and show better accuracy than those of the standard finite element method.     

Dynamic programming in minimum weight design of shells

In designing a structure for a given purpose, the designer should aim at the most economical design from many points of view. Minimum weight designs may not be the most economical ones. Nevertheless, such designs provide insights into other types of design and may pave ways for improvements in design procedures. Minimum weight designs of continuum structures may be formulated as numerical optimization problems using either the finite difference method or the finite element method, and the resulting minimization problem may be solved using techniques of numerical optimization. This paper, on the other hand, considers the formulation of the minimum weight design of a spherical cap as a dynamic programming problem and its solution. The paper also discusses the advantages and limitations of the approach.     

基于人工鱼群算法的结构模型修正与损伤检测

提出结构模型修正结构损伤检测的人工鱼群算法。将结构模型修正与结构损伤检测结构动力学逆问题转化为约束优化数学问题,并尝试用人工鱼群算法求解。介绍人工鱼群算法基本原理,定义关键参数并描述觅食、聚群、追尾及随机等行为;据模型修正原理利用结构损伤前后模态特性数据定义优化问题目标函数;通过两层刚架不同损伤工况数值仿真、三层框架试验数据验证方法的有效性。结果表明,基于人工鱼群算法的结构模型修正与损伤检测方法能有效修正结构有限元模型,在不同噪声水平及各种结构损伤工况下不仅能准确定位结构损伤且能精确识别损伤程度。     

考虑位移要求的大型三维连续体结构拓扑优化数值技术研究

大型复杂三维结构拓扑优化设计既具有理论意义,又具有重要的应用价值。基于等效转换的非奇异的结构优化模型,研究结构位移要求的最小结构重量设计问题。首先,介绍了位移约束的三维结构优化准则和公式。而后,为了提高拥有数万个单元以上的三维结构的计算效率,结合结构位移计算的迭代方法,在分析用于结构特性参数计算模型的基础上,建立了一套三维结构拓扑优化的求解策略和算法。最后,给出了几个典型和复杂的三维结构的拓扑优化设计算例。算例表明求解策略和算法是正确和有效的,且具有广泛的工程应用前景。     

Comparative study of granular solid-structure interaction in a uni-axial compression system

Interaction between granular solids and confining structures is an elementary problem encountered in subsurface structural design and bulk solids storing and handling. A classic scenario is uni-axial compression of granular solids in a deformable cylindrical container. Despite being apparently simple in loading condition, the understanding of this scenario remains limited, mainly due to complex interactive deformation between the two components via frictional interfaces. This paper comparatively examines such a uni-axial compression particulate system by a laboratory experiment and two different numerical approaches, namely, continuum finite element method (FEM) and linked discrete-finite element method (linked DEM-FEM). In the continuum FEM approach, two intendedly chosen simple material models, linear elastic and porous elastic models, are attempted. The comparative study reveals that the majority of resultant characteristics show satisfactory agreement amongst the numerical predictions and the experimental measurements. The simple elastic continuum FEM models can hence be a useful alternative in modelling such problems with mild structural flexibility under a monotonic loading scenario. However, precise prediction of some characteristics, such as lateral pressure ratio, may demand more elaborated material model or parameter selection. The enhancements needed for each numerical approach in order to achieve an improved result are further discussed.     

A decomposition procedure for large-scale optimum plastic design problems

A decomposition procedure is proposed in this paper for solving a class of large-scale optimum design problems for perfectly-plastic structures under several alternative loading conditions. The conventional finite element method is used to cast the problem into a finite dimensional constrained nonlinear programming problem. Structures of practically meaningful size and complexity tend to give rise to a large number of variables and constraints in the corresponding mathematical model. The difficulty is that the state-of-the-art mathematical programming theory does not provide reliable and efficient ways of solving large-scale constrained nonlinear programming problems. The natural idea to deal with the large-scale structural problem is somehow to decompose the problem into a collection of small-size problems each of which represents an analysis of the behaviour of each finite element under a single loading condition. This paper proposes one such way of decomposition based on duality theory and a recently developed iterative algorithm called the proximal point algorithm.     

Optimal layout design of three‐dimensional geometrically non‐linear structures using the element connectivity parameterization method

The topology design optimization of ‘three‐dimensional geometrically‐non‐linear’ continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization. To overcome these difficulties, the element connectivity parameterization (ECP) method with two implementation formulations is proposed. In ECP, structural layouts are represented by inter‐element connectivity, which is controlled by the stiffness of element‐connecting zero‐length links. Depending on the link location, ECP may be classified as an external ECP (E‐ECP) or an internal ECP (I‐ECP). In this paper, I‐ECP is newly developed to substantially enhance computational efficiency. The main idea in I‐ECP is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level. As for ECP implementation with commercial software, E‐ECP, developed earlier for two‐dimensional problems, is easier to use even for three‐dimensional problems because it requires only numerical analysis results for design sensitivity calculation. The characteristics of the I‐ECP and E‐ECP methods are compared, and these methods are validated with numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

多边形比例边界有限元非线性高效分析方法

比例边界有限元法作为一种高精度的半解析数值求解方法,特别适合于求解无限域与应力奇异性等问题,多边形比例边界单元在模拟裂纹扩展过程、处理局部网格重剖分等方面相较于有限单元法具有明显优势。目前,比例边界有限元法更多关注的是线弹性问题的求解,而非线性比例边界单元的研究则处于起步阶段。该文将高效的隔离非线性有限元法用于比例边界单元的非线性分析,提出了一种高效的隔离非线性比例边界有限元法。该方法认为每个边界线单元覆盖的区域为相互独立的扇形子单元,其形函数以及应变-位移矩阵可通过半解析的弹性解获得;每个扇形区的非线性应变场通过设置非线性应变插值点来表达,引入非线性本构关系即可实现多边形比例边界单元高效非线性分析。多边形比例边界单元的刚度通过集成每个扇形子单元的刚度获取,扇形子单元的刚度可采用高斯积分方案进行求解,其精度保持不变。由于引入了较多的非线性应变插值点,舒尔补矩阵维数较大,该文采用Woodbury近似法对隔离非线性比例边界单元的控制方程进行求解。该方法对大规模非线性问题的计算具有较高的计算效率,数值算例验证了算法的正确性以及高效性,将该方法进行推广,对实际工程分析具有重要意义。     

Recursive approach for random response analysis using non-orthogonal polynomial expansion

Using non-orthogonal polynomial expansions, a recursive approach is proposed for the random response analysis of structures under static loads involving random properties of materials, external loads, and structural geometries. In the present formulation, non-orthogonal polynomial expansions are utilized to express the unknown responses of random structural systems. Combining the high-order perturbation techniques and finite element method, a series of deterministic recursive equations is set up. The solutions of the recursive equations can be explicitly expressed through the adoption of special mathematical operators. Furthermore, the Galerkin method is utilized to modify the obtained coefficients for enhancing the convergence rate of computational outputs. In the post-processing of results, the first- and second-order statistical moments can be quickly obtained using the relationship matrix between the orthogonal and the non-orthogonal polynomials. Two linear static problems and a geometrical nonlinear problem are investigated as numerical examples in order to illustrate the performance of the proposed method. Computational results show that the proposed method speeds up the convergence rate and has the same accuracy as the spectral finite element method at a much lower computational cost, also, a comparison with the stochastic reduced basis method shows that the new method is effective for dealing with complex random problems.     

Meshfree natural vibration analysis of 2D structures

Determination of resonance frequencies and vibration modes of mechanical structures is one of the most important tasks in the product design procedure. The main goal of this paper is to describe a pioneering application of the solution structure method (SSM) to 2D structural natural vibration analysis problems and investigate the numerical properties of the method. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using analytical methods or traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward generalization to model vibrations of 3D structures.     

An adaptive procedure for eigenvalue problems using the hierarchical finite element method

An adaptive procedure for the solution of the generalized linear eigenvalue problem within the hierarchical finite element method is described. The problem of finding for a given discretization, an upper limit eigenvalue that is accurate within a prescribed tolerance is especially studied. An error estimator is presented and a recomputational scheme for improved solutions is proposed. A numerical example is included.