Material cloud method for topology optimization

Material cloud method (MCM), a new approach for topology optimization, is presented. In MCM, an optimal structure can be obtained by manipulating the sizes and positions of material clouds, which are material patches with finite sizes and constant material densities. The optimal distributions of material clouds can be obtained by MCM using fixed background finite element meshes. In the numerical analysis procedure, only active elements, where more than one material cloud is contained, are treated. Optimal material distribution can be element‐wise extracted from the distribution of material clouds. With MCM, an expansion–reduction procedure of design domain can be naturally realized through movements of material clouds, so that a true optimal solution can be found without any significant increase of computational costs. It is also shown that a clear material distribution with narrow region of intermediate density can be obtained with relatively fast convergence. Several numerical examples are shown. Some of the results are compared with those of the traditional density distribution method (DDM). Copyright © 2005 John Wiley & Sons, Ltd.     

A continuum approach for second-order shape design sensitivity of elastic solids with loaded boundaries

A second-order shape design sensitivity analysis (DSA) method applicable to the shape change on the loaded boundaries is derived for three-dimensional linear elastic solids using a continuum method with the material derivative. The continuum method is also used to derive mixed second-order variations of stress and displacement performance measures with respect to shape design variables and distribution of non-conservative traction loads, and also with respect to shape design variables and material properties. A shape design acceleration field is defined for the second-order shape design sensitivity. Both direct differentiation and hybrid methods are presented in this paper. A numerical method, which can be implemented using established finite element analysis (FEA) codes, is developed. The feasibility and accuracy of the proposed second-order shape DSA method has been demonstrated by solving a structural example-doubly curved arch dam.     

A robust preconditioning technique for the extended finite element method

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill‐conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill‐conditioned matrices. The method proposed here, however, provides well‐conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2. Copyright © 2010 John Wiley & Sons, Ltd.     

Treatment of discontinuity in the reproducing kernel element method

A discontinuous reproducing kernel element approximation is proposed in the case where weak discontinuity exists over an interface in the physical domain. The proposed method can effectively take care of the discontinuity of the derivative by truncating the window function and global partition polynomials. This new approximation keeps the advantage of both finite element methods and meshfree methods as in the reproducing kernel element method. The approximation has the interpolation property if the support of the window function is contained in the union of the elements associated with the corresponding node; therefore, the continuity of the primitive variables at nodes on the interface is ensured. Furthermore, it is smooth on each subregion (or each material) separated by the interface. The major advantage of the method is its simplicity in implementation and it is computationally efficient compared to other methods treating discontinuity. The convergence of the numerical solution is validated through calculations of some material discontinuity problems. Copyright © 2005 John Wiley & Sons, Ltd.     

周期性多材料结构稳态热传导拓扑优化设计

提出一种考虑周期性约束的多材料结构稳态热传导拓扑优化设计方法。针对多材料结构,提出基于有序有理近似材料属性模型（ordered rational approximation of material properties,Ordered-RAMP）的多材料插值模型。以结构散热弱度最小化为目标函数,体积为约束条件,将设计区域划分为有限个相同的子多材料区域。通过重新分配单元散热弱度基值,实现周期性几何约束,借助优化准则法推导设计变量的迭代格式。通过典型2D与3D数值算例,分析不同子区域个数对宏观结构与微观子区域多材料拓扑构型的影响。结果表明:所提方法可实现面向多材料结构的周期性微观构型设计,且各材料分布合理边界清晰,具有良好的稳健性;当子区域个数不同时,均可得到具有周期性的拓扑构型,且所获拓扑形式具有差异性。     

A nodal variable method of structural topology optimization based on Shepard interpolant

A method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Integrated layout design of multi‐component system

A new integrated layout optimization method is proposed here for the design of multi‐component systems. By introducing movable components into the design domain, the components layout and the supporting structural topology are optimized simultaneously. The developed design procedure mainly consists of three parts: (i) Introduction of non‐overlap constraints between components. The finite circle method (FCM) is used to avoid the components overlaps and also overlaps between components and the design domain boundaries. (ii) Layout optimization of the components and supporting structure. Locations and orientations of the components are assumed as geometrical design variables for the optimal placement while topology design variables of the supporting structure are defined by the density points. Meanwhile, embedded meshing techniques are developed to take into account the finite element mesh change caused by the component movements. (iii) Consistent material interpolation scheme between element stiffness and inertial load. The commonly used solid isotropic material with penalization model is improved to avoid the singularity of localized deformation in the presence of design dependent loading when the element stiffness and the involved inertial load are weakened by the element material removal. Finally, to validate the proposed design procedure, a variety of multi‐component system layout design problems are tested and solved on account of inertia loads and gravity center position constraint. Solutions are compared with traditional topology designs without component. Copyright © 2008 John Wiley & Sons, Ltd.     

Topology optimization using a continuous density field and adaptive mesh refinement

A new method of topology optimization is introduced in which a continuous material field is combined with adaptive mesh refinement. Using a continuous material field with different analysis and design meshes allows the method to produce optimal designs that are free of numerical artifacts like checkerboard patterns and material islands. Adaptive mesh refinement is then applied to both meshes to precisely locate the optimal boundary of the final structure. A Helmholtz‐type density filter is used to prevent the appearance of small topological features as the mesh refinement proceeds. Results are presented for several test problems, including problems with geometrically complex domain boundaries.     

Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation

In this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one‐dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Comparison of high‐order curved finite elements

Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points on the accuracy of the computation is also studied. Two‐dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS‐enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element method (FEM). Moreover, NEFEM outperforms Cartesian FEM and p‐FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Mixed finite element formulations in the time domain for solution of dynamic problems

This paper presents a two-field mixed finite element formulation in the time domain. Several strategies are presented by which the variational formulations are produced. One strategy is based on Hamilton's law, and is suitable for problems where the mechanical energy of the system is given.Another strategy uses the Galerkin method, which is suitable when the equations of motion are given. This strategy can either be based on a continuous or discontinuous approximation of the physical variables in time. All formulations were analyzed, and their characteristics in terms of truncation error (phase and amplitude) and stability are presented. This analysis was verified by numerical results. Comparison of many formulations leads to the conclusion that discontinuous mixed finite element method in the time domain is superior to all others in terms of stability and accuracy.     

Adaptive meshless local maximum-entropy finite element method for convection–diffusion problems

In this paper, a meshless local maximum-entropy finite element method (LME-FEM) is proposed to solve 1D Poisson equation and steady state convection–diffusion problems at various Peclet numbers in both 1D and 2D. By using local maximum-entropy (LME) approximation scheme to construct the element shape functions in the formulation of finite element method (FEM), additional nodes can be introduced within element without any mesh refinement to increase the accuracy of numerical approximation of unknown function, which procedure is similar to conventional p-refinement but without increasing the element connectivity to avoid the high conditioning matrix. The resulted LME-FEM preserves several significant characteristics of conventional FEM such as Kronecker-delta property on element vertices, partition of unity of shape function and exact reproduction of constant and linear functions. Furthermore, according to the essential properties of LME approximation scheme, nodes can be introduced in an arbitrary way and the $C^0$ continuity of the shape function along element edge is kept at the same time. No transition element is needed to connect elements of different orders. The property of arbitrary local refinement makes LME-FEM be a numerical method that can adaptively solve the numerical solutions of various problems where troublesome local mesh refinement is in general necessary to obtain reasonable solutions. Several numerical examples with dramatically varying solutions are presented to test the capability of the current method. The numerical results show that LME-FEM can obtain much better and stable solutions than conventional FEM with linear element.     

Numerical discretization of stationary random processes

The increasing interest of the research community to the probabilistic analysis concerning the civil structures with space-variant properties points out the problem of achieving a reliable discretization of random processes (or random fields in a multi-dimensional domain). Given a discretization method, a continuous random process is approximated by a finite set of random variables. Its dimension affects significantly the accuracy of the approximation, in terms of the relevant properties of the continuous random process under investigation. The paper presents a discretization procedure based on the truncated Karhunen–Loève series expansion and the finite element method. The objective is to link in a rational way the number of random variables involved in the approximation to a quantitative measure of the discretization accuracy. The finite element method is applied to evaluate the terms of the series expansion when a closed form expression is not available. An iterative refinement of the finite element mesh is proposed in this paper, leading to an accurate random process discretization. The technique is tested with respect to the exponential covariance function, that enables a comparison with analytical expressions of the approximated properties of the random process. Then, the procedure is applied to the square exponential covariance functions, which is one of the most used covariance models in the structural engineering field. The comparison of the adaptive refinement of the discretization with a non-adaptive procedure and with the wavelet Galerkin approach allows to demonstrate the computational efficiency of the proposal within the framework of the Karhunen–Loève series expansion. A comparison with the Expansion Optimal Linear Estimation (EOLE) method is performed in terms of efficiency of the discretization strategy.     

Scale‐related topology optimization of cellular materials and structures

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two‐scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so‐called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale‐related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

Slope constrained topology optimization

The problem of minimum compliance topology optimization of an elastic continuum is considered. A general continuous density–energy relation is assumed, including variable thickness sheet models and artificial power laws. To ensure existence of solutions, the design set is restricted by enforcing pointwise bounds on the density slopes. A finite element discretization procedure is described, and a proof of convergence of finite element solutions to exact solutions is given, as well as numerical examples obtained by a continuation/SLP (sequential linear programming) method. The convergence proof implies that checkerboard patterns and other numerical anomalies will not be present, or at least, that they can be made arbitrarily weak. © 1998 John Wiley & Sons, Ltd.     

Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4‐node tetrahedral and 8‐node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three‐dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.     

Using finite element analysis/optimum design in the determination of elastic constants by vibration testing

Abstract

Combining vibration testing and numerical methods is a potential alternative approach for determining elastic constants of materials because of nondestructive testing, single testing, and producing average properties. In order to simplify the modeling processes and to reduce complicated derivations in the numerical method, the combination of finite element analysis and optimum design is adopted in this work to inversely calculate the elastic constants. In this method, the effects of some parameters, including the finite element mesh, the mode number, constraints on state variables, the optimization method, and initial values and loops, are discussed. It is proved that the present method with 10×10 mesh, 6 vibration modes, 5% constraint on state variables, and subproblem approximation can obtain accurate results. Average values can be used to avoid the scatter in results due to different initial values, and do‐loops may not be necessary. It is also clearly demonstrated that the present method can correctly extract 2 or 5 elastic constants for isotropic or orthotropic materials.     

Probabilistic analysis of concrete cracking using neural networks and random fields

In this paper an algorithm for the probabilistic analysis of concrete structures is proposed which considers material uncertainties and failure due to cracking. The fluctuations of the material parameters are modeled by means of random fields and the cracking process is represented by a discrete approach using a coupled meshless and finite element discretization. In order to analyze the complex behavior of these nonlinear systems with low numerical costs a neural network approximation of the performance functions is realized. As neural network input parameters the important random variables of the random field in the uncorrelated Gaussian space are used and the output values are the interesting response quantities such as deformation and load capacities. The neural network approximation is based on a stochastic training which uses wide spanned Latin hypercube sampling to generate the training samples. This ensures a high quality approximation over the whole domain investigated, even in regions with very small probability.     

Effective 3D failure simulations by combining the advantages of embedded Strong Discontinuity Approaches and classical interface elements

An efficient numerical framework suitable for three-dimensional analyses of brittle material failure is presented. The proposed model is based on an (embedded) Strong Discontinuity Approach (SDA). Hence, the deformation mapping is elementwise additively decomposed into a conforming, continuous part and an enhanced part associated with the kinematics induced by material failure. To overcome locking effects and to provide a continuous crack path approximation, the approach is extended and combined with advantages known from classical interface elements. More precisely, several discontinuities each of them being parallel to a facet of the respective finite element are considered. By doing so, crack path continuity is automatically fulfilled and no tracking algorithm is necessary. However, though this idea is similar to that of interface elements, the novel SDA is strictly local (finite element level) and thus, it does not require any modification of the global data structure, e.g. no duplication of nodes. An additional positive feature of the advocated finite element formulation is that it leads to a symmetric tangent matrix. It is shown that several simultaneously active discontinuities in each finite element are required to capture highly localized material failure. The performance and predictive ability of the model are demonstrated by means of two benchmark examples.