A conformal decomposition finite element method for arbitrary discontinuities on moving interfaces

Interface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities. The conformal decomposition FEM decomposes multimaterial elements of a non‐conforming background mesh into sub‐elements that conform to material interfaces captured using a level set method. As the interface evolves, interfacial nodes move, and background nodes may change material. The present work describes approaches for handling moving interfaces in the context of the conformal decomposition FEM for both weakly and strongly discontinuous fields. Dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first‐order and second‐order time integration methods. The moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one‐dimensional and two‐dimensional test problems, while achieving the expected second‐order error convergence rate in space and time. Copyright © 2014 John Wiley & Sons, Ltd.     

Refined non‐conforming triangular elements for analysis of shell structures

Based on the refined non‐conforming element method, simple flat triangular elements with standard nodal displacement parameters are proposed for the analysis of shell structures. For ensuring the convergence of the elements a new coupled continuity condition at the inter‐element has been established in a weaker form. A common displacement for the inter‐element, an explicit expression of refined constant strain matrix, and an adjustable constant are introduced into the formulation, in which the coupled continuity requirement at the inter‐element is satisfied in the average sense. The non‐conforming displacement function of the well‐known triangular plate element BCIZ [1] and the membrane displacement of the constant strain triangular element CST [2] are employed to derive the refined flat shell elements RTS15, and the refined flat shell elements RTS18 is derived by using the element BCIZ and the Allman's triangular plane element [3] with the drilling degrees of freedom. A simple reduced higher‐order membrane strain matrix is proposed to avoid membrane locking of the element RTS18. An alternative new reduced higher‐order strain matrix method is developed to improve the accuracy of the elements RTS15 and RTS18. Numerical examples are given to show that the present methods have improved the accuracy of the shell analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

The finite element tearing and interconnecting (FETI) method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second‐order elasticity problems. In the boundary element tearing and interconnecting (BETI) method, a direct extension of the FETI algorithm to the BEM, the symmetric Galerkin BEM formulation, is used to obtain symmetric system matrices, making possible to apply the same FETI conjugate gradient solver. In this work, we propose a new BETI variant labeled nsBETI that allows to couple substructures modeled with the FEM and/or non‐symmetrical BEM formulations. The method connects non‐matching BEM and FEM subdomains using localized Lagrange multipliers and solves the associated non‐symmetrical flexibility equations with a Bi‐CGstab iterative algorithm. Scalability issues of nsBETI in BEM–BEM and combined BEM–FEM coupled problems are also investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

The abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.     

Direct modification for non‐conforming elements with drilling DOF

A unified approach to eliminate the undesirable lockings in distorted meshes for both non‐conforming quadrilateral membrane and hexahedral elements with drilling (or rotational) degrees of freedom is presented. The direct modification scheme is utilized in the formulation of non‐conforming modes to improve the general behaviour of isoparametric‐based elements. It is shown that the direct modification is very effective in eliminating the locking and this improvement of element behaviour may be doubled if the selective integration scheme is used simultaneously. To verify the validity of elements formulated by the proposed schemes and to evaluate their effectiveness, several numerical tests are carried out. The combined use of additional non‐conforming modes with the direct modification method and the selective integration technique plays an important role to improve the behaviour of elements, especially for distorted mesh cases. Test results also show that the results for both non‐conforming quadrilateral membrane and hexahedral elements obtained by the proposed schemes are equally satisfactory. Copyright © 2002 John Wiley & Sons, Ltd.     

Topology optimization using non‐conforming finite elements: three‐dimensional case

As in the case of two‐dimensional topology design optimization, numerical instability problems similar to the formation of two‐dimensional checkerboard patterns occur if the standard eight‐node conforming brick element is used. Motivated by the recent success of the two‐dimensional non‐conforming elements in completely eliminating checkerboard patterns, we aim at investigating the performance of three‐dimensional non‐conforming elements in controlling the patterns that are estimated overly stiff by the brick elements. To this end, we will investigate how accurately the non‐conforming elements estimate the stiffness of the patterns. The stiffness estimation is based on the homogenization method by assuming the periodicity of the patterns. To verify the superior performance of the elements, we consider three‐dimensional compliance minimization and compliant mechanism design problems and compare the results by the non‐conforming element and the standard 8‐node conforming brick element. Copyright © 2005 John Wiley & Sons, Ltd.     

Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non‐conforming four‐node finite elements are employed. Since the convergence of the non‐conforming finite element is independent of the Lamé parameters, the stiffness of the non‐conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard‐free property of the non‐conforming element in topology optimization problems and verify it with three typical optimization examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Structural‐acoustic coupling on non‐conforming meshes with quadratic shape functions

Fully coupled finite element/boundary element models are a popular choice when modelling structures that are submerged in heavy fluids. To achieve coupling of subdomains with non‐conforming discretizations at their common interface, the coupling conditions are usually formulated in a weak sense. The coupling matrices are evaluated by integrating products of piecewise polynomials on independent meshes. The case of interfacing elements with linear shape functions on unrelated meshes has been well covered in the literature. This paper presents a solution to the problem of evaluating the coupling matrix for interfacing elements with quadratic shape functions on unrelated meshes. The isoparametric finite elements have eight nodes (Serendipity) and the discontinuous boundary elements have nine nodes (Lagrange). Results using linear and quadratic shape functions on conforming and non‐conforming meshes are compared for an example of a fluid‐loaded point‐excited sphere. It is shown that the coupling error decreases when quadratic shape functions are used. Copyright © 2012 John Wiley & Sons, Ltd.     

Conforming local meshfree method

This work is motivated by the current numerical limitation in multiscale simulation of ductile fracture processes at scale down to the microstructure size and aims to overcome the difficulties in 3D complicated mesh generation and locally extremely large strain analysis (local mesh distortion). The proposed ‘conforming local meshfree approximation’ directly and exactly satisfies displacement compatibility on a non‐conforming assembly mesh. Local meshfree nodes, which can be freely placed and move on a finite element mesh, describe local large deformation. The improved accuracy on non‐conforming mesh, the exactness in geometry representation on a structured mesh, and the good tolerance to mesh distortion are demonstrated by numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

Adaptive modeling of damage growth using a coupled FEM/BEM approach

We propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non‐linear damage growth is accurately accounted by a local FEM mesh. An integral‐type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non‐linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

An adaptive simplex cut‐cell method for high‐order discontinuous Galerkin discretizations of conjugate heat transfer problems

In this paper, we present a solution framework for high‐order discretizations of conjugate heat transfer problems on non‐body‐conforming meshes. The framework consists of and leverages recent developments in discontinuous Galerkin discretization, simplex cut‐cell techniques, and anisotropic output‐based adaptation. With the cut‐cell technique, the mesh generation process is completely decoupled from the interface definitions. In addition, the adaptive scheme combined with the discontinuous Galerkin discretization automatically adjusts the mesh in each sub‐domain and achieves high‐order accuracy in outputs of interest. We demonstrate the solution framework through several multi‐domained conjugate heat transfer problems consisting of laminar and turbulent flows, curved geometry, and highly coupled heat transfer regions. The combination of these attributes yield nonintuitive coupled interactions between fluid and solid domains, which can be difficult to capture with user‐generated meshes. Copyright © 2016 John Wiley & Sons, Ltd.     

Implicit boundary method for finite element analysis using non‐conforming mesh or grid

In the finite element method (FEM), a mesh is used for representing the geometry of the analysis and for representing the test and trial functions by piece‐wise interpolation. Recently, analysis techniques that use structured grids have been developed to avoid the need for a conforming mesh. The boundaries of the analysis domain are represented using implicit equations while a structured grid is used to interpolate functions. Such a method for analysis using structured grids is presented here in which the analysis domain is constructed by Boolean combination of step functions. Implicit equations of the boundary are used in the construction of trial and test functions such that essential boundary conditions are guaranteed to be satisfied. Furthermore, these functions are constructed such that internal elements, through which no boundary passes, have the same stiffness matrix. This approach has been applied to solve linear elastostatic problems and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the FEM in quality but is less computationally expensive for dense mesh/grid and avoids the need for a conforming mesh. Copyright © 2007 John Wiley & Sons, Ltd.     

Dual boundary‐element method: Simple error estimator and adaptivity

This paper is concerned with the effective numerical implementation of the adaptive dual boundary‐element method (DBEM), for two‐dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single‐region analysis. Taking advantage on the use of non‐conforming parametric boundary‐elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright © 2011 John Wiley & Sons, Ltd.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

An extended finite element method approach for structural‐acoustic problems involving immersed structures at arbitrary positions

Noise reduction for passengers' comfort in transport industry is now an important constraint to be taken into account during the design process. This process involves to study several configurations of the structures immersed in a given acoustic cavity in the context of an optimization, uncertainty, or reliability study for instance. The finite element method may be used to model this coupled fluid–structure problem but needs an interface conforming mesh for each studied configuration that may become time consuming. This work aims at avoiding this remeshing step by using noncompatible meshes between the fluid and the structures. The immersed structures are supposed to be thin shells and are localized in the fluid domain by a signed distance level‐set. To take into account the pressure discontinuity from one side of the structures to the other one, the fluid pressure approximation is enriched according to the structures positions by a Heaviside function using a partition of unity strategy (extended finite element method). The same fluid mesh of the empty cavity is then used during the whole parametric study. The method is implemented for a three‐dimensional fluid and tested on academic examples before being applied to an industrial‐like case. Copyright © 2012 John Wiley & Sons, Ltd.     

Lower‐dimensional interface elements with local enrichment: application to coupled hydro‐mechanical problems in discretely fractured porous media

In this study, we develop lower‐dimensional interface elements to represent preexisting fractures in rock material, focusing on finite element analysis of coupled hydro‐mechanical problems in discrete fractures–porous media systems. The method adopts local enrichment approximations for a discontinuous displacement and a fracture relative displacement function. Multiple and intersected fractures can be treated with the new scheme. Moreover, the method requires less mesh dependencies for accurate finiteelement approximations compared with the conventional interface element method. In particular, for coupled problems, the method allows for the use of a single mesh for both mechanical and other related processes such as flow and transport. For verification purposes, several numerical examples are examined in detail. Application to a coupled hydro‐mechanical problem is demonstrated with fluid injection into a single fracture. The numerical examples prove that the proposed method produces results in strong agreement with reference solutions. Copyright © 2012 John Wiley & Sons, Ltd.