A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

A coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain. Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite element analyses of wave propagations due to a high‐speed train across bridges

The finite element mesh of the unbounded wave problem often contains a large number of degrees of freedom. Moreover, the wave reflection along the mesh boundary cannot be avoided if only the traditional time‐domain finite element method (FEM) is used. Thus, the main purpose of this study is to overcome those drawbacks and discuss how to use the three‐dimensional FEM to simulate the soil vibration due to a moving high‐speed train across bridges. A number of finite element analyses (FEA) including the isolation schemes of in‐filled and open trenches were performed. The appropriate mesh dimension in the direction of the railroad is suggested and validated. Moreover, the 3‐node wheel element for simulating the moving train and the least‐squares method for finding foundation properties are proposed. Since the finite element mesh contains a large number of degrees of freedom, the conjugated gradient method with the SSOR preconditioning scheme is recommended. In conclusion, this study indicates that the difficulty of the FEA for unbounded problems can be overcome efficiently even by using a personal computer. Copyright © 2002 John Wiley & Sons, Ltd.     

Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.     

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

The edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

A fully symmetric multi‐zone Galerkin boundary element method

This paper examines the efficient integration of a Symmetric Galerkin Boundary Element Analysis (SGBEA) method with multi‐zone resulting in a fully symmetric Galerkin multi‐zone formulation. In a previous approach, a Galerkin multi‐zone method was developed where the interfacial nodes are assigned degrees of freedom globally so that the displacement and traction continuity across the zonal interfaces are addressed directly. However, the method was only block symmetric. In the present paper, two new approaches are derived. In the first approach, the degrees of freedom for a particular zone are assigned locally, independent of the other zones. The usual linear set of equations, from the symmetric Galerkin approach, are augmented with an additional set of equations generated by the Galerkin form of hypersingular boundary integrals along the interfaces. Zonal continuity is imposed externally through Lagrange's constraints. This approach is also only block symmetric. The second approach derived from the first, uses the continuity constraints at the zonal assembly level to achieve full symmetry. These methods are compared to collocation multi‐zone and an earlier formulation, on two elasticity problems from the literature. It was found that the second method is much faster than the collocation method for medium to large scale problems, primarily due to its complete symmetry. It is also observed that these methods spend marginally more time on integration than the previous Galerkin multi‐zone method but are better suited to parallel processing. Copyright © 1999 John Wiley & Sons, Ltd.     

Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

In this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress‐driven and strain‐driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite‐element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill–Mandel principle of macro‐homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated ‘forces’ that are conjugate to the macroscopic strain ‘displacements’. In contrast to most homogenization schemes, the second Piola–Kirchhoff stress and Green–Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola–Kirchhoff stress and the deformation gradient is discussed in the Appendix. Copyright © 2015 John Wiley & Sons, Ltd.     

Thermoelastic stress field in a piecewise homogeneous domain under non‐uniform temperature using a coupled boundary and finite element method

This study concerns the development of a coupled finite element–boundary element analysis method for the solution of thermoelastic stresses in a domain composed of dissimilar materials with geometric discontinuities. The continuity of displacement and traction components is enforced directly along the interfaces between different material regions of the domain. The presence of material and geometric discontinuities are included in the formulation explicitly. The unknown interface traction components are expressed in terms of unknown interface displacement components by using the boundary element method for each material region of the domain. Enforcing the continuity conditions leads to a final system of equations containing unknown interface displacement components only. With the solution of interface displacement components, each region has a complete set of boundary conditions, thus leading to the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, stresses from specific BEM regions are first expressed in terms of interface displacements, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of FEM regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

An interface‐enriched generalized FEM for problems with discontinuous gradient fields

A new generalized FEM is introduced for solving problems with discontinuous gradient fields. The method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges. The proposed approach has several advantages over conventional generalized FEM formulations, such as a lower computational cost, easier implementation, and straightforward handling of Dirichlet boundary conditions. A detailed convergence study of the proposed method and a comparison with the standard FEM are presented for heat transfer problems. The method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard FEM with conforming meshes. Various application problems are presented, including the conjugate heat transfer problem encountered in microvascular materials. Copyright © 2011 John Wiley & Sons, Ltd.     

Convergence analysis of a parallel interfield method for heterogeneous simulations with dynamic substructuring

In order to predict the dynamic response of a complex system decomposed by computational or physical considerations, partitioned procedures of coupled dynamical systems are needed. This paper presents the convergence analysis of a novel parallel interfield procedure for time‐integrating heterogeneous (numerical/physical) subsystems typical of hardware‐in‐the‐loop and pseudo‐dynamic tests. The partitioned method is an extension of the method originally proposed by Gravouil and Combescure which utilizes a domain decomposition enforcing the continuity of the velocity at interfaces. In particular, the merits of the new method which can couple arbitrary Newmark schemes with different time steps in different subdomains and advance all the substructure states simultaneously are analysed in terms of accuracy and stability. All theoretical results are derived for single‐ and two‐degrees‐of‐freedom systems, as a multi‐degree‐of‐freedom system is too difficult to analyse mathematically. However, the insight gained from the analysis of these coupled problems and the conclusions drawn are confirmed by means of the numerical simulation on a four‐degrees‐of‐freedom system. Copyright © 2008 John Wiley & Sons, Ltd.     

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

High‐order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high‐order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial‐boundary value problems, eigenvalue problems, and high‐order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high‐order differential equations and time‐dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high‐order accuracy, while maintaining the same or similar stability conditions of the standard high‐order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non‐standard high‐order methods is also considered. Copyright © 2008 John Wiley & Sons, Ltd.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

Adaptive atomistic‐to‐continuum modeling of propagating defects

An adaptive atomistic‐to‐continuum method is presented for modeling the propagation of material defects. This method extends the bridging domain method to allow the atomic domain to dynamically conform to the evolving defect regions during a simulation, without introducing spurious oscillations and without requiring mesh refinement. The atomic domain expands as defects approach the bridging domain method coupling domain by fine graining nearby finite elements into equivalent atomistic subdomains. Additional algorithms coarse grain portions of the atomic domain to the continuum scale, reducing the degrees of freedom, when the atomic displacements in a subdomain can be approximated by FEM or extended FEM elements to within a certain homogeneity tolerance. The extended FEM approximations are created by fitting the broken inter‐atomic bonds of fractured surfaces and dislocation slip planes. Because atomic degrees of freedom are maintained only where needed for each timestep, the solution retains the advantages of multiscale modeling, with a reduced computational cost compared with other multiscale methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes are presented. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favourable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three‐node triangular or four‐node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behaviour for a set of example problems. Published in 2000 by John Wiley & Sons, Ltd.