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Showing papers in "International Journal for Numerical Methods in Engineering in 2010"


Journal ArticleDOI
TL;DR: In this article, a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, developed incremental variational principles and considering their numerical implementations by multi-field finite element methods is presented.
Abstract: The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

1,555 citations


Journal ArticleDOI
TL;DR: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented in this article, which enables accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements.
Abstract: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

1,228 citations


Journal ArticleDOI
TL;DR: It was found that the distribution of the ϕp values tends to lower values as the dimensionality is increased and the proposed translational propagation algorithm represents a computationally attractive strategy to obtain near optimum LHDs up to medium dimensions.
Abstract: CITATION: Viana, F. A. C., Venter, G. & Balabanov, V. 2010. An algorithm for fast optimal Latin hypercube design of experiments. International Journal for Numerical Methods in Engineering, 82(2):135-156, doi:10.1002/nme.2750.

287 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the weak enforcement of Dirichlet boundary conditions for B-spline basis functions, with application to both second-and fourth-order problems.
Abstract: A key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.

274 citations


Journal ArticleDOI
TL;DR: In this paper, a generalized gradient smoothing technique is used for a unified formulation of a wide class of compatible and incompatible methods, which can have special properties including softened behavior, upper bounds and ultra accuracy.
Abstract: This paper introduces a G space theory and a weakened weak form (W2) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W2 formulation works for both finite element method settings and mesh-free settings, and W2 models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for W2 formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the W2 formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ‘close-to-exact’ stiffness, upper bounds and ultra accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

263 citations


Journal ArticleDOI
TL;DR: In this paper, a rate-dependent plastic material model is proposed and demonstrated to accurately reproduce the experimental results for Taylor impact tests over a wide range of impact velocities, and the resulting model retains the advantages of the peridynamic formulation regarding discontinuities while allowing greater generality in material response than was previously possible.
Abstract: Peridynamics is a continuum reformulation of the standard theory of solid mechanics. Unlike the partial differential equations of the standard theory, the basic equations of peridynamics are applicable even when cracks and other singularities appear in the deformation field. The assumptions in the original peridynamic theory resulted in severe restrictions on the types of material response that could be modeled, including a limitation on the Poisson ratio. Recent theoretical developments have shown promise for overcoming these limitations, but have not previously incorporated rate dependence and have not been demonstrated in realistic applications. In this paper, a new method for implementing a rate-dependent plastic material within a peridynamic numerical model is proposed and demonstrated. The resulting material model implementation is fitted to rate-dependent test data on 6061-T6 aluminum alloy. It is shown that with this material model, the peridynamic method accurately reproduces the experimental results for Taylor impact tests over a wide range of impact velocities. The resulting model retains the advantages of the peridynamic formulation regarding discontinuities while allowing greater generality in material response than was previously possible. Copyright © 2009 John Wiley & Sons, Ltd.

255 citations


Journal ArticleDOI
TL;DR: This paper presents a method that separates as much as possible the generation and evaluation of the basis functions from the analysis, resulting in a formulation that can be implemented within the traditional structure of a finite element program but that permits the use of arbitrary sets of basis functions that are defined only through the input file.
Abstract: Many of the formulations of cm-rent research interest, including iosogeometric methods and the extended finite element method, use nontraditional basis functions. Some, such as subdivision surfaces, may not have convenient analytical representations. The concept of an element, if appropriate at all, no longer coincides with the traditional definition. Developing a new software for each new class of basis functions is a large research burden, especially, if the problems involve large deformations, non-linear materials, and contact. The objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis, resulting in a formulation that can be implemented within the traditional structure of a finite clement program but that permits the use of arbitrary sets of basis functions that are defined only through the input file. Elements ranging from a traditional linear four-node tetrahedron through a higher-order element combining XFEM and isogeometric analysis may be specified entirely through an input file without any additional programming. Examples of this framework to applications with Lagrange elements, isogeometric elements, and XFEM basis functions for fracture are presented.

223 citations


Journal ArticleDOI
TL;DR: In this article, a bilinear W 2 formulation for solid mechanics problems is proposed, which is based on the G space theory and is shown to be spatially stable and convergent to exact solutions.
Abstract: In part I of this paper, we have established the G space theory and fundamentals for W 2 formulation. Part II focuses on the applications of the G space theory to formulate W 2 models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W 2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W 2 models including the SFEM, NS-FEM, ES-FEM, NS-PIM, ES-PIM, and CS-PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W 2 models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W 2 models including compatible and incompatible cases. We shall see that the G space theory and the W 2 forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case.

211 citations


Journal ArticleDOI
TL;DR: A numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons, which can be used as software libraries where numerical integration within planar polygons is required.
Abstract: In this paper, we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons. These quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required. We have used this algorithm for the construction of symmetric and non-symmetric quadrature rules over convex and concave polygons. While in the case of symmetric quadratures our results are comparable to available rules, the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions. To demonstrate the efficiency of the new quadrature rules, we have tested them for the integration of rational polygonal shape functions over a regular hexagon. For a relative error of 10−8 in the computation of stiffness matrix entries, one needs at least 198 evaluation points when the region is partitioned, whereas 85 points suffice with our quadrature rule. Copyright © 2009 John Wiley & Sons, Ltd.

181 citations


Journal ArticleDOI
TL;DR: Polygonal meshes constructed from Voronoi tessellations are considered, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities.
Abstract: In topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian-type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one-node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh-dependent sub-optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. Copyright © 2009 John Wiley & Sons, Ltd.

171 citations


Journal ArticleDOI
TL;DR: In this article, an optimal transportation mesh-free (OTM) method was developed for simulating general solid and fluid flows, including fluid-structure interaction, which combines concepts from optimal transportation theory with material-point sampling and max-ent meshfree interpolation.
Abstract: We develop an optimal transportation meshfree (OTM) method for simulating general solid and fluid flows, including fluid–structure interaction. The method combines concepts from optimal transportation theory with material-point sampling and max-ent meshfree interpolation. The proposed OTM method generalizes the Benamou–Brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior. The OTM method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities. The OTM method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems. We illustrate the range and scope of the method by means of two examples of application: the bouncing of a gas-filled balloon off a rigid wall; and the classical Taylor-anvil benchmark test extended to the hypervelocity range.

Journal ArticleDOI
TL;DR: In this paper, a primal-dual active set strategy for direct constraint enforcement is presented for 3D frictionless contact based on a dual mortar formulation and using a primal dual active set.
Abstract: In this paper, an approach for three-dimensional frictionless contact based on a dual mortar formulation and using a primal-dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher-order interpolation as well. The study builds on previous work by the authors for two-dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi-smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis.

Journal ArticleDOI
TL;DR: In this paper, a monolithic procedure employing a unified domain rather than separated domains is proposed for topology optimization in the steady-state fluid-structure interaction (FSI) problem, where the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor.
Abstract: This paper outlines a new procedure for topology optimization in the steady-state fluid-structure interaction (FSI) problem. A review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics (hereafter, the fluid and structural equations, respectively) and in imposing coupling boundary conditions between the separated fluid and solid domains. To overcome these difficulties, we propose an alternative monolithic procedure employing a unified domain rather than separated domains, which is not computationally efficient. In the proposed analysis procedure, the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor. For the coupling boundary conditions, the divergence of the pressure and the Darcy damping force are inserted into the solid and fluid equations, respectively. The proposed method is validated in several benchmark analysis problems. Topology optimization in the FSI problem is then made possible by interpolating Young's modulus, the fluid pressure of the modified solid equation, and the inverse permeability from the damping force with respect to the design variables.

Journal ArticleDOI
TL;DR: In this paper, an explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane, and coupled with a boundary integral method to solve for the internal and external Stokes flows.
Abstract: We introduce a new numerical method to model the fluid–structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in-plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions. Copyright © 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a computational multiscale framework is proposed that incorporates microstructural behaviour in a macro-scale discrete fracture model, and homogenization procedures are derived for both adhesive and cohesive failure on the macroscale and are implemented in an FE 2 -setting.
Abstract: A computational multiscale framework is proposed that incorporates microstructural behaviour in a macroscale discrete fracture model. Homogenization procedures are derived for both adhesive and cohesive failure on the macroscale and are implemented in an FE 2 -setting. The most important feature of the homogenization procedure is that it implicitly defines a traction-opening relation for the macroscale fracture model. The representativeness of the micro models is studied using a one-dimensional example, which shows that in the softening regime the proposed multiscale scheme behaves different from a bulk homogenization scheme. These results are also observed in a numerical simulation for a micro model with a periodic microstructure. Numerical simulations further demonstrate the applicability of the method.

Journal ArticleDOI
TL;DR: A general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S‐FEM models is presented and a general error bound is established.
Abstract: Incorporating the strain smoothing technique of meshfree methods into the standard finite element method (FEM), Liu et al. have recently proposed a series of smoothed finite element methods (S-FEM) for solid mechanics problems. In these S-FEM models, the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements, nodes, edges or faces, and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. We present in this paper a general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models. First, an assumed strain field derived from the Hellinger–Reissner variational principle is shown to be identical to the smoothed strain field used in the S-FEM models. We then define a smoothing projection operator to modify the compatible strain field and show a set of properties. We next establish a general error bound of the S-FEM models. Some numerical examples are given to verify the theoretical properties established. Copyright © 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, an edge-based smoothed finite element method (ES-FEM) using three-node linear triangular elements was proposed to significantly improve the accuracy and convergence rate of the standard finite element formulation for static, free and forced vibration analyses of solids.
Abstract: An edge-based smoothed finite element method (ES-FEM) using three-node linear triangular elements was recently proposed to significantly improve the accuracy and convergence rate of the standard finite element formulation for static, free and forced vibration analyses of solids. In this paper, ES-FEM is further extended for limit and shakedown analyses of structures. A primal-dual algorithm based upon the von Mises yield criterion and a non-linear optimization procedure is used to compute both the upper and lower bounds of the plastic collapse limit and the shakedown limit. In the ES-FEM, compatible strains are smoothed over the smoothing domains associated with edges of elements. Using constant smoothing function, only one Gaussian point is required for each smoothing domain ensuring that the total number of variables in the resulting optimization problem is kept to a minimum compared with standard finite element formulation. Three benchmark problems are presented to show the stability and accuracy of solutions obtained by the present method.

Journal ArticleDOI
TL;DR: In this article, the displacement is reconstructed by minimizing the least-squared errors between measured and approximated acceleration within a finite time interval, and an overlapping time window is introduced to improve the accuracy of the reconstructed displacement.
Abstract: This paper presents a new class of displacement reconstruction scheme using only acceleration measured from a structure. For a given set of acceleration data, the reconstruction problem is formulated as a boundary value problem in which the acceleration is approximated by the second-order central finite difference of displacement. The displacement is reconstructed by minimizing the least-squared errors between measured and approximated acceleration within a finite time interval. An overlapping time window is introduced to improve the accuracy of the reconstructed displacement. The displacement reconstruction problem becomes ill-posed because the boundary conditions at both ends of each time window are not known a priori. Furthermore, random noise in measured acceleration causes physically inadmissible errors in the reconstructed displacement. A Tikhonov regularization scheme is adopted to alleviate the ill-posedness. It is shown that the proposed method is equivalent to an FIR filter designed in the time domain. The fundamental characteristics of the proposed method are presented in the frequency domain using the transfer function and the accuracy function. The validity of the proposed method is demonstrated by a numerical example, a laboratory experiment and a field test. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a multiscale aggregating discontinuity (MAD) method for coarse graining of micro-cracks to the macro-scale was further developed, and three new features were introduced: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods of recovering macrocracks with variable crack opening.
Abstract: A method for coarse graining of microcrack growth to the macroscale through the multiscale aggregating discontinuity (MAD) method is further developed. Three new features are: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods for recovering macrocracks with variable crack opening. Unlike in the original MAD method, ellipticity is not retained at the macroscale in the bulk material, but we show that the element stiffness of the bulk material is positive definite. Several examples with comparisons with direct numerical simulations are given to demonstrate the effectiveness of the method. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi-material problem show that the proposed integration technique can be easily integrated in any existing code and yields accurate results.
Abstract: Partition of unity methods, such as the extended finite element method, allows discontinuities to be simulated independently of the mesh (Int. J. Numer. Meth. Engng. 1999; 45:601-620). This eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re-meshing, as the discontinuity evolves. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted. In this paper, we use a simple integration technique, proposed for polygonal domains (Int. J. Nuttier Meth. Engng 2009; 80(1):103-134. DOI: 10.1002/nme.2589) to suppress the need for element subdivision. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi-material problem show that the proposed method yields accurate results. Owing to its simplicity, the proposed integration technique can be easily integrated in any existing code. Copyright (C) 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: The methods key idea is the use of an additional stress field as the constraining Lagrange multiplier function, which allows the straight-forward application of state-of-the-art iterative solvers, like Algebraic Multigrid techniques.
Abstract: This paper presents a new approach for imposing Dirichlet conditions weakly on non-fitting finite element meshes. Such conditions, also called embedded Dirichlet conditions, are typically, but not exclusively, encountered when prescribing Dirichlet conditions in the context of the eXtended Finite Element Method (XFEM). The methods key idea is the use of an additional stress field as the constraining Lagrange multiplier function. The resulting mixed/hybrid formulation is applicable to 1D, 2D and 3D problems. The method does not require stabilization for the Lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level. Furthermore, only non- zero diagonal-terms are present in the tangent stiffness, which allows the straight-forward application of state-of-the-art iterative solvers, like Algebraic Multigrid (AMG) techniques. Within this paper, the method is applied to the linear momentum equation of an elastic continuum and to the transient, incompressible Navier-Stokes equations. Steady and unsteady benchmark computations show excellent agreement with reference values. The general formulation presented in this paper can also be applied to other continuous field problems.

Journal ArticleDOI
TL;DR: In this article, a single step algorithm based on a mixed optical/mechanical cost function was proposed to identify a non-linear consitutive law, where no boundary conditions are needed.
Abstract: Constitutive parameter identification has been greatly improved by the achievement of full-field measurements. In this context, noise sensitivity has been shown to be of great importance. It is crucial to incorporate noise sensitivity minimization in the design of robust identification procedures. In this paper, we investigate noise sensitivity reduction techniques for constitutive parameter identification based on Finite Element Model Updating. After examining the existing strategies, we propose a single step algorithm based on a mixed optical/mechanical cost function. The key point of this novel procedure is that no boundary conditions are needed. A first example on a real case illustrates the advantages of the proposed methodology in terms of noise sensitivity. A second example shows its capabilities to identify a non-linear consitutive law. Copyright © 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)-scale problems is proposed, which enables accurate modeling of three-dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts.
Abstract: This paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)-scale problems. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of three-dimensional cracks, while the global problem addresses the macro-scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp-GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three-dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three-dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the authors leverage properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization, which can be used to reduce the number of independent design variables without significantly restricting the design space.
Abstract: Topology optimization methodologies typically use the same discretization for the design variable and analysis meshes. Analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense. This paper proposes leveraging properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization. HPM projects independent design variables onto element space over a prescribed length scale. A single design variable therefore influences several elements, creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space. The algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses. The technique is demonstrated on minimum compliance (maximum stiffness) problems solved using continuous optimization and genetic algorithms. For the former, the proposed algorithm typically identifies solutions having objective functions within 1% of those found using full design variable fields. Computational savings are minor to moderate for the minimum compliance formulation with a single constraint, and are substantial for formulations having many local constraints. When using genetic algorithms, solutions are consistently obtained on mesh resolutions that were previously considered intractable. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a computational homogenization technique for thin-structured sheets is proposed, based on the computational homogeneization concepts for first-and second-order continua, for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element.
Abstract: In this paper, a computational homogenization technique for thin-structured sheets is proposed, based on the computational homogenization concepts for first- and second-order continua. The actual three-dimensional (3D) heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element (RVE), incorporating the full thickness of the sheet and an in-plane representative cell of the macroscopic structure. At an in-plane integration point of the macroscopic shell, the generalized strains, i.e. the membrane deformation and the curvature, are used to formulate the boundary conditions for the microscale RVE problem. At the RVE scale, all microstructural constituents are modeled as an ordinary 3D continuum, described by the standard equilibrium and the constitutive equations. Upon proper averaging of the RVE response, the macroscopic generalized stress and the moment resultants are obtained. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. From a macroscopic point of view, a (numerical) generalized stress-strain constitutive response at every macroscopic in-plane integration point is obtained. Additionally, the simultaneously resolved microscale RVE local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design.

Journal ArticleDOI
TL;DR: In this article, an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation is proposed.
Abstract: This paper proposes an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation. The macroscopic elasticity tensor is estimated by a standard multi-scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element. The algorithm is of simple computational implementation. In particular, it does not require artificial algorithmic parameters or strategies. This is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept. This concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems. The effectiveness of the proposed methodology is illustrated in a set of finite element-based numerical examples.Copyright © 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, an explicit-explicit staggered time-integration algorithm and an implicit-implicit counterpart are presented for the solution of non-linear transient fluid-structure interaction problems in the Arbitrary Lagrangian-Eulerian (ALE) setting.
Abstract: An explicit–explicit staggered time-integration algorithm and an implicit–explicit counterpart are presented for the solution of non-linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two-stage explicit Runge–Kutta and three-point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second-order time-accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time-integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second-order time-accuracy are verified for sample application problems. Their potential for the solution of highly non-linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration-free time-integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a new numerical procedure for kinematic limit analysis is presented, which incorporates the cell-based smoothed finite element method with second-order cone programming and results in an efficient method that can provide accurate solutions with minimal computational effort.
Abstract: This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged.

Journal ArticleDOI
TL;DR: In this article, a new structural optimization method based on the level set method using a new geometry-based re-initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level-set function are performed using the Finite Element Method.
Abstract: Structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function, and updated by solving the so-called Hamilton–Jacobi equation based on a Eulerian coordinate system. These new methods can allow topological alterations, such as the number of holes, during the optimization process whereas the boundaries of the target structure are clearly defined. However, the re-initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures. In this paper, we propose a new structural optimization method based on the level set method using a new geometry-based re-initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the Finite Element Method. The stiffness maximization, eigenfrequency maximization, and eigenfrequency matching problems are considered as optimization problems. Several design examples are presented to confirm the usefulness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a singular node-based smoothed finite element method (NS-FEM) is proposed to obtain the upper bound solutions for fracture problems, where the strain smoothing technique over the smoothing domains (SDs) associated with nodes is performed, which leads to the line integrations using only the shape function values along the boundaries of the SDs.
Abstract: It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node-based smoothed finite element method (NS-FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS-FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five-node singular crack tip element is used within the framework of NS-FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix-mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS-FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.