Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Combined three‐dimensional lattice Boltzmann method and discrete element method for modelling fluid–particle interactions with experimental assessment

A general algorithmic framework is established in this paper for numerical simulations of three‐dimensional fluid–particle interaction problems with a large number of moving particles in turbulent flows using a combined lattice Boltzmann method (LBM) and discrete element method (DEM). In this approach, the fluid field is solved by the extended three‐dimensional LBM with the incorporation of the Smagorinsky turbulence model, while particle interactions are modelled by the DEM. The hydrodynamic interactions between fluid and particles are realized through the extension of an existing two‐dimensional fluid–particle hydrodynamic interaction scheme. The main computational aspects comprise the lattice Boltzmann formulation for the solution of fluid flows, the incorporation of a large eddy simulation‐based turbulence model within the framework of the three‐dimensional LBM for turbulent flows, the moving boundary condition for hydrodynamic interactions between fluid and moving particles, and the discrete element modelling of particle‐particle interactions. To assess the solution accuracy of the proposed approach, a much simplified laboratory model of vacuum dredging systems for mineral recovery is employed. The numerical results are compared with the experimental data available. It shows that the overall correspondence between numerical results and experimental measurements is good and thus indicates, to a certain extent, the solution accuracy of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient augmented finite element method for arbitrary cracking and crack interaction in solids

This paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary, multiple intra‐elemental discontinuities in 2D solids. The augmented finite element method (A‐FEM) employs four internal nodes to account for the crack displacements due to an intra‐elemental weak or strong discontinuity, and it permits repeated elemental augmentation to include multiple interactive cracks. It thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity, and to multiple interactive cohesive cracks, all within a single bilinear element that employs standard external nodal DoFs only. A novel elemental condensation procedure has been developed to solve the internal nodal DoFs as functions of the external nodal DoFs for any irreversible, piece‐wise linear cohesive laws. It leads to a fully condensed elemental equilibrium equation with mathematical exactness, eliminating the need for nonlinear equilibrium iterations at elemental level. The new A‐FEM's high‐fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous, and heterogeneous solids have been demonstrated through several challenging numerical examples. It is shown that the proposed A‐FEM, empowered by the new elemental condensation procedure, is numerically very efficient, accurate, and robust. Copyright © 2014 John Wiley & Sons, Ltd.     

Improving multiresolution topology optimization via multiple discretizations

Unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization, this paper proposes a framework for improving multiresolution topology optimization (iMTOP) via multiple distinct discretizations for: (1) finite elements; (2) design variables; and (3) density. This approach leads to high fidelity resolution with a relatively low computational cost. In addition, an adaptive multiresolution topology optimization (AMTOP) procedure is introduced, which consists of selective adjustment and refinement of design variable and density fields. Various two‐dimensional and three‐dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element‐based approach. Copyright © 2012 John Wiley & Sons, Ltd.     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

Uncertain static plane stress analysis with interval fields

Uncertain static plane stress analysis of continuous structure involving interval fields is investigated in this study. Unlike traditional interval analysis of discrete structure, the interval field is adopted to model the uncertainty, as well as the dependency between the physical locations and degrees of variability, of all interval system parameters presented in the continuous structures. By implementing the flexibility properties of some common structural elements, a new computational scheme is proposed to reformulate the uncertain static plane stress analysis with interval fields into standard mathematical programming problems. Consequently, feasible upper and lower bounds of structural responses can be effectively yet efficiently determined. In addition, the proposed method is adequate to deal with situations involving one‐dimensional and two‐dimensional interval fields, which enhances the pertinence of the proposed approach by incorporating both discrete and continuous structures. In addition, the proposed computational scheme is able to establish the realizations of the uncertain parameters causing the extreme structural responses at zero computational cost. The applicability and credibility of the established computational framework are rigorously justified by various numerical investigations. Copyright © 2016 John Wiley & Sons, Ltd.     

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

We present a computational framework for the simulation of J₂‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.     

A stable and optimally convergent LaTIn‐CutFEM algorithm for multiple unilateral contact problems

In this paper, we propose a novel unfitted finite element method for the simulation of multiple body contact. The computational mesh is generated independently of the geometry of the interacting solids, which can be arbitrarily complex. The key novelty of the approach is the combination of elements of the CutFEM technology, namely, the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty‐type regularisation of discrete operators and the LaTIn hybrid‐mixed formulation of complex interface conditions. Furthermore, the novel P1‐P1 discretisation scheme that we propose for the unfitted LaTIn solver is shown to be stable, robust, and optimally convergent with mesh refinement. Finally, the paper introduces a high‐performance 3D level set/CutFEM framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries, with more than 2 bodies interacting at a single point.     

A novel discrete element method based on the distance potential for arbitrary 2D convex elements

A new 2‐dimensional discrete element method, which is able to simulate a system involving a large number of arbitrary convex elements, is proposed. In this approach, a novel distance potential function is defined using a normalized format of the penetrated distance between contact couples, while a holonomic and precise algorithm for contact interaction is established, accounting for the influence of the tangential contact force. Furthermore, the new contact detection algorithm is well suited for nonuniform blocks unlike the common no binary search method that requires uniform elements. The proposed method retains the merit of the combined finite‐discrete element method and avoids its deficiencies. Compared with the existing finite‐discrete element method, the distance potential function has a clear physical meaning, where the calculation of contact interaction avoids the influence of the element shape. Accordingly, the new method completely gets rid of the restraint of uniform element type and can be applied to arbitrary convex elements. The new method is validated with well‐known benchmark examples, and the results are in very good agreement with existing experimental measurement and analytical solutions. Finally, the proposed method is applied to simulate the Tangjiashan landslide.     

Robust and direct evaluation of J2 in linear elastic fracture mechanics with the X‐FEM

The aim of the present paper is to study the accuracy and the robustness of the evaluation of J_k‐integrals in linear elastic fracture mechanics using the extended finite element method (X‐FEM) approach. X‐FEM is a numerical method based on the partition of unity framework that allows the representation of discontinuity surfaces such as cracks, material inclusions or holes without meshing them explicitly. The main focus in this contribution is to compare various approaches for the numerical evaluation of the J₂‐integral. These approaches have been proposed in the context of both classical and enriched finite elements. However, their convergence and the robustness have not yet been studied, which are the goals of this contribution. It is shown that the approaches that were used previously within the enriched finite element context do not converge numerically and that this convergence can be recovered with an improved strategy that is proposed in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Discrete gradient method over polygon mesh

This paper presents a discrete method over domains originally discretized by polygons including triangle, quadrilateral, and general n‐sided polygon elements. In this method, the domain is re‐partitioned into node‐based cells. At each node, the gradient of a physical variable is approximated using a linearly exact discrete operator that involves a set of neighboring nodes. The discrete gradient is subsequently substituted into a weak form to yield a nodal‐integration Galerkin formulation. A unified geometric approach is provided for constructing the gradient operators over an arbitrary polygon mesh. The method does not introduce continuous approximation of the unknown variable; therefore, the numerical computation is very simple. The linear displacement patch test is satisfied by construction. Numerical tests show that the method has comparable accuracy and convergence rate as the displacement finite element method. Examples are also included to illustrate the ability to resist numerical locking in the incompressibility limit and the thin‐element limit. Copyright © 2008 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

A‐scalability and an integrated computational technology and framework for non‐linear structural dynamics. Part 2: Implementation aspects and parallel performance results

An integrated framework and computational technology is described that addresses the issues to foster absolute scalability (A‐scalability) of the entire transient duration of the simulations of implicit non‐linear structural dynamics of large scale practical applications on a large number of parallel processors. Whereas the theoretical developments and parallel formulations were presented in Part 1, the implementation, validation and parallel performance assessments and results are presented here in Part 2 of the paper. Relatively simple numerical examples involving large deformation and elastic and elastoplastic non‐linear dynamic behaviour are first presented via the proposed framework for demonstrating the comparative accuracy of methods in comparison to available experimental results and/or results available in the literature. For practical geometrically complex meshes, the A‐scalability of non‐linear implicit dynamic computations is then illustrated by employing scalable optimal dissipative zero‐order displacement and velocity overshoot behaviour time operators which are a subset of the generalized framework in conjunction with numerically scalable spatial domain decomposition methods and scalable graph partitioning techniques. The constant run times of the entire simulation of ‘fixed‐memory‐use‐per‐processor’ scaling of complex finite element mesh geometries is demonstrated for large scale problems and large processor counts on at least 1024 processors. Copyright © 2003 John Wiley & Sons, Ltd.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

基于离散元和有限差分耦合算法的崩塌体和结构碰撞预测分析

采用离散元和有限差分耦合计算方法,对龙洞子崩塌挡墙结构碰撞进行了预测分析。构建了3种崩塌挡墙结构模型,并进行了数值模拟。崩塌体由颗粒模拟,挡墙分别由刚性墙、颗粒、FLAC中的实体单元模拟。基于离散元和有限差分耦合算法的数值模拟,分析了不同模型的挡墙动力响应和变形。通过数值模拟结果的对比分析,选出了最优的结构模型和参数,为有效防御崩塌的防护结构设计提供理论依据。     

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Large‐scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency

A rigorous computational framework for the dimensional reduction of discrete, high‐fidelity, nonlinear, finite element structural dynamics models is presented. It is based on the pre‐computation of solution snapshots, their compression into a reduced‐order basis, and the Galerkin projection of the given discrete high‐dimensional model onto this basis. To this effect, this framework distinguishes between vector‐valued displacements and manifold‐valued finite rotations. To minimize computational complexity, it also differentiates between the cases of constant and configuration‐dependent mass matrices. Like most projection‐based nonlinear model reduction methods, however, its computational efficiency hinges not only on the ability of the constructed reduced‐order basis to capture the dominant features of the solution of interest but also on the ability of this framework to compute fast and accurate approximations of the projection onto a subspace of tangent matrices and/or force vectors. The computation of the latter approximations is often referred to in the literature as hyper reduction. Hence, this paper also presents the energy‐conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi‐discrete), nonlinear, finite element structural dynamics models. Based on mesh sampling and the principle of virtual work, ECSW is natural for finite element computations and preserves an important energetic aspect of the high‐dimensional finite element model to be reduced. Equipped with this hyper reduction procedure, the aforementioned Galerkin projection framework is first demonstrated for several academic but challenging problems. Then, its potential for the effective solution of real problems is highlighted with the realistic simulation of the transient response of a vehicle to an underbody blast event. For this problem, the proposed nonlinear model reduction framework reduces the CPU time required by a typical high‐dimensional model by up to four orders of magnitude while maintaining a good level of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.     

基于GPU并行的锥体导管架平台结构冰激振动DEM-FEM耦合分析

该文为分析海冰与锥体海洋平台的相互作用,采用离散元（DEM）-有限元（FEM）耦合方法建立冰激海洋平台结构的耦合模型。通过具有粘结-破碎性能的球体离散单元对海冰的漂移及破碎现象进行计算,海洋平台锥体部分采用平板型壳单元构造,其整体构架及锥体内部的加劲肋采用梁单元构造,即建立壳单元与梁单元组合的锥体海洋平台有限元模型。为提高DEM-FEM耦合算法的计算规模和效率,发展了离散单元与平板型壳单元接触算法及GPU并行环境下参数传递算法。基于此耦合模型分别讨论了平台结构的冰载荷、冰激振动以及锥体应力分布,并与相关实测数据进行对比,为寒区锥体海洋平台的结构设计提供有益的参考。     

Discrete direct and adjoint sensitivity analysis for arbitrary Mach number flows

Parallel discrete direct and adjoint sensitivity analysis capabilities are developed for arbitrary Mach flows on mixed‐element unstructured grids. The discrete direct and adjoint methods need a consistent and complete linearization of the flow‐solver to obtain accurate derivatives. The discontinuous nature of the commonly used unstructured flux‐limiters, like Barth–Jespersen and Venkatakrishnan, make them unsuitable for sensitivity analysis. A modification is proposed to make these limiters piecewise continuous and numerically differentiable, without compromising the monotonicity conditions. An improved version of Symmetric Gauss–Seidel that significantly reduces the computational cost is implemented. A distributed‐memory message passing model is employed for the parallelization of sensitivity analysis solver. These algorithms are implemented within a three‐dimensional unstructured grid framework and results are presented for inviscid, laminar and turbulent flows. Copyright © 2005 John Wiley & Sons, Ltd.