A localized mixed‐hybrid method for imposing interfacial constraints in the extended finite element method (XFEM)

The paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily‐shaped material interfaces using an unfitted background mesh. The use of Lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations. In a defined region in the vicinity of the interface, the original problem is re‐stated in its auxiliary formulation. The availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form. The contribution discusses the weak imposition of Dirichlet‐ and Neumann‐type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation. The properties of the proposed approach applied to two‐dimensional linear scalar‐ and vector‐valued elliptic problems are investigated by studying the convergence behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

Embedded interfaces by polytope FEM

In this paper, the Polytope Finite Element Method is employed to model an embedded interface through the body, independent of the background FEM mesh. The elements that are crossed by the embedded interface are decomposed into new polytope elements which have some nodes on the interface line. The interface introduces discontinuity into the primary variable (strong) or into its derivatives (weak). Both strong and weak discontinuities are studied by the proposed method through different numerical examples including fracture problems with traction‐free and cohesive cracks, and heat conduction problems with Dirichlet and Dirichlet–Neumann types of boundary conditions on the embedded interface. For traction‐free cracks which have tip singularity, the nodes near the crack tip are enriched with the singular functions through the eXtended Finite Element Method. The concept of Natural Element Coordinates (NECs) is invoked to drive shape functions for the produced polytopes. A simple treatment is proposed for concave polytopes produced by a kinked interface and also for locating crack tip inside an element prior to using the singularity enrichment. The proposed method pursues some implementational details of eXtended/Generalized Finite Element Methods for interfaces. But here the additional DOFs are constructed on the interface lines in contrast to X/G‐FEM, which attach enriched DOFs to the previously existed nodes. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

An adaptive interface‐enriched generalized FEM for the treatment of problems with curved interfaces

An adaptive refinement scheme is presented to reduce the geometry discretization error and provide higher‐order enrichment functions for the interface‐enriched generalized FEM. The proposed method relies on the h‐adaptive and p‐adaptive refinement techniques to reduce the discrepancy between the exact and discretized geometries of curved material interfaces. A thorough discussion is provided on identifying the appropriate level of the refinement for curved interfaces based on the size of the elements of the background mesh. Varied techniques are then studied for selecting the quasi‐optimal location of interface nodes to obtain a more accurate approximation of the interface geometry. We also discuss different approaches for creating the integration sub‐elements and evaluating the corresponding enrichment functions together with their impact on the performance and computational cost of higher‐order enrichments. Several examples are presented to demonstrate the application of the adaptive interface‐enriched generalized FEM for modeling thermo‐mechanical problems with intricate geometries. The accuracy and convergence rate of the method are also studied in these example problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Conformal higher‐order remeshing schemes for implicitly defined interface problems

A new higher‐order accurate method is proposed that combines the advantages of the classical p‐version of the FEM on body‐fitted meshes with embedded domain methods. A background mesh composed by higher‐order Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher‐order accurate sub‐elements is realized. Therefore, the zero level sets are detected and meshed in a first step, which is called reconstruction. Then, based on the topological situation in the cut element, higher‐order sub‐elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub‐elements are not always well‐shaped. Copyright © 2016 John Wiley & Sons, Ltd.     

Interface handling for three‐dimensional higher‐order XFEM‐computations in fluid–structure interaction

Three‐dimensional higher‐order eXtended finite element method (XFEM)‐computations still pose challenging computational geometry problems especially for moving interfaces. This paper provides a method for the localization of a higher‐order interface finite element (FE) mesh in an underlying three‐dimensional higher‐order FE mesh. Additionally, it demonstrates, how a subtetrahedralization of an intersected element can be obtained, which preserves the possibly curved interface and allows therefore exact numerical integration. The proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their ‘eXtended axis‐aligned bounding boxes’. The intersection method is applied to a highly reduced number of intersection candidates. The resulting linearized interface is used as input for an elementwise constrained Delaunay tetrahedralization, which computes an appropriate subdivision for each intersected element. The curved interface is recovered from the linearized interface in the last step. The output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element. Application of the interface algorithm currently concentrates on fluid–structure interaction problems on low‐order and higher‐order FE meshes, which may be composed of any arbitrary element types such as hexahedra, tetrahedra, wedges, etc. Nevertheless, other XFEM‐problems with explicitly given interfaces or discontinuities may be tackled in addition. Multiple structures and interfaces per intersected element can be handled without any additional difficulties. Several parallelization strategies exist depending on the desired domain decomposition approach. Numerical test cases including various geometrical exceptions demonstrate the accuracy, robustness and efficiency of the interface handling. Copyright © 2009 John Wiley & Sons, Ltd.     

The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.     

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

This paper presents a FEM with mesh‐separation‐based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution‐dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi‐analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack‐tip adaptive mesh refinement strategy to improve the accuracy of crack‐tip fields. Both the discontinuities due to intra‐element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A 2‐D meshless model for jointed rock structures

According to the characteristic structural features of jointed rock structures, a meshless model is proposed for the mechanics analysis of jointed rock structures based on the moving least‐squares interpolants. In this model, a jointed rock structure is regarded as a system of relatively intact rock blocks connected by joints or planes of discontinuity; these rock blocks are modelled by general shaped anisotropic blocks while these joints and planes of discontinuity are modelled by interfaces. The displacement field of each block is constructed by the moving least‐squares interpolants with an array of points distributed in the block. To deal with the discontinuities of rock structures, the displacement fields are constructed to be discontinuous between blocks. The displacement fields and their gradients are continuous in each block, hence no post processing is required for the output of strains and stresses. The finite element mesh is totally unnecessary, so the time‐consuming mesh generation is avoided. The rate of convergence can exceed that of finite elements significantly, and a high resolution of localized steep gradients can be achieved. Furthermore, the discontinuities of rock structures are also fully taken into consideration. The present method is developed for two‐dimensional linear elastic analysis of jointed rock structures, and can be extended to three‐dimensional and non‐linear analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

On the application of higher‐order elements in the hierarchical interface‐enriched finite element method

This article introduces a new algorithm for evaluating enrichment functions in the higher‐order hierarchical interface‐enriched finite element method (HIFEM), which enables the fully mesh‐independent simulation of multiphase problems with intricate morphologies. The proposed hierarchical enrichment technique can accurately capture gradient discontinuities along materials interfaces that are in close proximity, in contact, and even intersecting with one another using nonconforming finite element meshes for discretizing the problem. We study different approaches for creating higher‐order HIFEM enrichments corresponding to six‐node triangular elements and analyze the advantages and shortcomings of each approach. The preferred method, which yields the lowest computational cost and highest accuracy, relies on a special mapping between the local and global coordinate systems for evaluating enrichment functions. A comprehensive convergence study is presented to show that this method yields similar convergence rate and precision as those of the standard FEM with conforming meshes. Finally, we demonstrate the application of the higher‐order HIFEM for simulating the thermal and deformation responses of several materials systems and engineering problems with complex geometries. Copyright © 2015 John Wiley & Sons, Ltd.     

Arbitrary placement of local meshes in a global mesh by the interface‐element method (IEM)

A new method is proposed to place local meshes in a global mesh with the aid of the interface‐element method (IEM). The interface‐elements use moving least‐square (MLS)‐based shape functions to join partitioned finite‐element domains with non‐matching interfaces. The supports of nodes are defined to satisfy the continuity condition on the interfaces by introducing pseudonodes on the boundaries of interface regions. Particularly, the weight functions of nodes on the boundaries of interface regions span only neighbouring nodes, ensuring that the resulting shape functions are identical to those of adjoining finite‐elements. The completeness of the shape functions of the interface‐elements up to the order of basis provides a reasonable transfer of strain fields through the non‐matching interfaces between partitioned domains. Taking these great advantages of the IEM, local meshes can be easily inserted at arbitrary places in a global mesh. Several numerical examples show the effectiveness of this technique for modelling of local regions in a global domain. Copyright © 2003 John Wiley & Sons, Ltd.     

Higher‐order extended FEM for weak discontinuities – level set representation,quadrature and application to magneto‐mechanical problems

In this article, we present the application of bilinear and biquadratic extended FEM (XFEM) formulations to model weak discontinuities in magnetic and coupled magneto‐mechanical boundary value problems. For properly resolving the location of curved interfaces and the discontinuous physical behaviour, the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher‐order XFEM formulations. In order to reduce the complexity of the representation of curved interfaces, an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed. Integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains. Eventually, a coupled magneto‐mechanical demonstration problem is modelled and solved by XFEM. For demonstration purposes, a magneto‐mechanical coupling due to magnetic stresses is considered. Errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface. Copyright © 2012 John Wiley & Sons, Ltd.     

Treatment of contact separation in Eulerian high‐speed multimaterial dynamic simulations

A frictionless contact separation treatment in a sharp‐interface Eulerian framework is presented to handle the general situation of high‐speed impact and separation of materials. The algorithm has been developed for an established Eulerian‐based Cartesian grid multimaterial flow code in which the interfaces are tracked in a sharp manner using a standard narrow‐band level set approach. Boundary conditions have been applied using a modified ghost fluid method for elasto‐plastic materials. The sharp‐interface treatment maintains the distinct interacting interfaces without smearing the contact zone while also removing the difficulties associated with Lagrangian moving mesh entities in contact‐separation situations. The algorithm has been tested and verified against experimental and numerical results for three different problems in the high strain rate regime, which involve contact, separation and sliding of materials. Copyright © 2014 John Wiley & Sons, Ltd.     

Moving particle finite element method

This paper presents the fundamental concepts behind the moving particle finite element method, which combines salient features of finite element and meshfree methods. The proposed method alleviates certain problems that plague meshfree techniques, such as essential boundary condition enforcement and the use of a separate background mesh to integrate the weak form. The method is illustrated via two‐dimensional linear elastic problems. Numerical examples are provided to show the capability of the method in benchmark problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub?domain smoothed Galerkin method is proposed to integrate the advantages of mesh?free Galerkin method and FEM. Arbitrarily shaped sub?domains are predefined in problems domain with mesh?free nodes. In each sub?domain, based on mesh?free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high?order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub?domain by dividing the sub?domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub?domains. The mesh?free properties of Galerkin method are retained in each sub?domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub?domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.     

Multimaterial multijoint topology optimization

In this paper, a methodology that solves multimaterial topology optimization problems while also optimizing the quantity and type of joints between dissimilar materials is proposed. Multimaterial topology optimization has become a popular design optimization technique since the enhanced design freedom typically leads to superior solutions; however, the conventional assumption that all elements are perfectly fused together as a single piece limits the usefulness of the approach since the mutual dependency between optimal multimaterial geometry and optimal joint design is not properly accounted for. The proposed methodology uses an effective decomposition approach to both determine the optimal topology of a structure using multiple materials and the optimal joint design using multiple joint types. By decomposing the problem into two smaller subproblems, gradient‐based optimization techniques can be used and large models that cannot be solved with nongradient approaches can be solved. Moreover, since the joining interfaces are interpreted directly from multimaterial topology optimization results, the shape of the joining interfaces and the quantity of joints connecting dissimilar materials do not need to be defined a priori. Three numerical examples, which demonstrate how the methodology optimizes the geometry of a multimaterial structure for both compliance and cost of joining, are presented.     

An embedded statistical method for coupling molecular dynamics and finite element analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with finite element (FE) nodes at their common interface, necessarily requiring that the FE mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modelling to two‐dimensional material domains due to difficulties in simulating full three‐dimensional material processes. In the present work, a new approach to MD–FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and FE nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. Thus, the method lends itself for use with any FEM or MD code. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three‐dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied. Published in 2009 by John Wiley & Sons, Ltd.     

The extended finite element method for fracture in composite materials

Methods for treating fracture in composite material by the extended finite element method with meshes that are independent of matrix/fiber interfaces and crack morphology are described. All discontinuities and near‐tip enrichments are modeled using the framework of local partition of unity. Level sets are used to describe the geometry of the interfaces and cracks so that no explicit representation of either the cracks or the material interfaces are needed. Both full 12 function enrichments and approximate enrichments for bimaterial crack tips are employed. A technique to correct the approximation in blending elements is used to improve the accuracy. Several numerical results for both two‐dimensional and three‐dimensional examples illustrate the versatility of the technique. The results clearly demonstrate that interface enrichment is sufficient to model the correct mechanics of an interface crack. Copyright © 2008 John Wiley & Sons, Ltd.     

Failure of heterogeneous materials: 3D meso‐scale FE models with embedded discontinuities

We present a meso‐scale model for failure of heterogeneous quasi‐brittle materials. The model problem of heterogeneous materials that is addressed in detail is based on two‐phase 3D representation of reinforced heterogeneous materials, such as concrete, where the inclusions are melt within the matrix. The quasi‐brittle failure mechanisms are described by the spatial truss representation, which is defined by the chosen Voronoi mesh. In order to explicitly incorporate heterogeneities with no need to change this mesh, some bar elements are cut by the phase‐interface and must be split into two parts. Any such element is enhanced using both weak and strong discontinuities, based upon the Incompatible Mode Method. Furthermore, a dedicated operator split solution procedure is proposed to keep local any additional computation on elements with embedded discontinuities. The results for several numerical simulations are presented to illustrate the capabilities of the proposed model to provide an excellent representation of failure mechanisms for any different macroscopic loading path. Copyright © 2010 John Wiley & Sons, Ltd.