A hybrid multiscale finite element/peridynamics method

A hybrid method is presented that uses a representative volume element-based multiscale finite element technique combined with a peridynamics method for modeling fracture surfaces. The hybrid method dynamically switches from finite element computations to peridynamics based on a damage criterion defined on the peridynamics grid, which is coincident with the nodes of the finite element mesh. Nodal forces are either computed by the finite element method or peridynamics, as appropriate. The multiscale finite element method used here is a representative volume element-based approach so that inhomogeneous local scale material properties can be derived using homogenization. In addition, automatic cohesive zone insertion is used at the local scale to model fracture initiation. Results demonstrate that local scale flaw distributions can alter fracture patterns and initiation times, and the use of cohesive zone insertion can improve accuracy of crack paths.     

A least‐squares coupling method between a finite element code and a discrete element code

This paper presents a coupling method between a discrete element code CeaMka3D and a finite element code Sem. The coupling is based on a least‐squares method, which adds terms of forces to finite element code and imposes the velocity at coupling particles. For each coupling face, a small linear system with a constant matrix is solved. This method remains conservative in energy and shows good results in applications. Copyright © 2014 John Wiley & Sons, Ltd.     

The quadratic programming approach to the finite element method

A finite element approximation technique is described in which the potential energy function is minimized subject to liner constraints. The constraints require satisfaction of all kinematic boundary conditions and interelement continuity conditions. An example, indicating the efficiency of the proposed method, is presented.     

Modelling cohesive crack growth using a two-step finite element-scaled boundary finite element coupled method

A two-step method, coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM), is developed in this paper for modelling cohesive crack growth in quasi-brittle normal-sized structures such as concrete beams. In the first step, the crack trajectory is fully automatically predicted by a recently-developed simple remeshing procedure using the SBFEM based on the linear elastic fracture mechanics theory. In the second step, interfacial finite elements with tension-softening constitutive laws are inserted into the crack path to model gradual energy dissipation in the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the proposed method. The numerical results demonstrate that this two-step SBFEM-FEM coupled method can predict both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for crack growth problems with strong snap-back phenomenon. The effects of the tensile strength, the mode-I and mode-II fracture energies on the predicted load-displacement relations are also discussed.     

Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM-FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K_I ? 0 criterion is objective with respect to the crack incremental length.     

A coupled analysis method for structures with independently modelled finite element subdomains

A new method for analysing plate and shell structures with two or more independently modelled finite element subdomains is presented, assessed, and demonstrated. This method provides a means of coupling local and global finite element models whose nodes do not coincide along their common interface. In general, the method provides a means of coupling structural components (e.g., wing and fuselage) which may have been modelled by different analysts. In both cases, the need for transition modelling, which is often tedious and complicated, is eliminated. The coupling is accomplished through an interface for which three formulations are considered and presented. These formulations are: collocation, discrete least-squares, and hybrid variational. Several benchmark problems are analysed and it is shown that the hybrid variational formulation provides the most accurate solutions.     

Monte Carlo method implemented in a finite element code with application to dynamic vacuum in particle accelerators

Modern particle accelerators require UHV conditions during their operation. In the accelerating cavities, breakdowns can occur, releasing large amount of gas into the vacuum chamber. To determine the pressure profile along the cavity as a function of time, the time-dependent behaviour of the gas has to be simulated. To do that, it is useful to apply accurate three-dimensional method, such as Test Particles Monte Carlo. In this paper, a time-dependent Test Particles Monte Carlo is used. It has been implemented in a Finite Element code, CASTEM. The principle is to track a sample of molecules during time. The complex geometry of the cavities can be created either in the FE code or in a CAD software (CATIA in our case). The interface between the two softwares to export the geometry from CATIA to CASTEM is given. The algorithm of particle tracking for collisionless flow in the FE code is shown. Thermal outgassing, pumping surfaces and electron and/or ion stimulated desorption can all be generated as well as different surface properties. The method is used to determine the pressure profile after breakdown in the Compact Linear Collider accelerating structures. Preliminary results are presented in this paper.     

Parallel adaptive finite element computations with hierarchical preconditioning

A parallel implementation of an adaptive finite element program is treated which is characterized by an underlying parallel dynamic data structure based on linked lists and tree structures. In conjunction with a conjugate gradient solver an efficient methodology for treating adaptive finite element systems is shown. This is achieved by preconditioning using hierarchical bases with and without a coarse grid solver and by new methods of quasi-optimal load balancing. The different levels of nested meshes needed for preconditioning are governed either by global or by adaptive refinements. A termination algorithm based on the vector method is implemented for the non deterministic adaptive mesh refinement procedure. The problems concerning load balancing due to adaptive refinement are solved by a dynamic load balancing for the nodes.This work has been supported by Deutsche Forschungsgemeinschaft under grant no. Ste238/26.     

Parallel simulations of three-dimensional cracks using the generalized finite element method

This paper presents a parallel generalized finite element method (GFEM) that uses customized enrichment functions for applications where limited a priori knowledge about the solution is available. The procedure involves the parallel solution of local boundary value problems using boundary conditions from a coarse global problem. The local solutions are in turn used to enrich the global solution space using the partition of unity methodology. The parallel computation of local solutions can be implemented using a single pair of scatter–gather communications. Several numerical experiments demonstrate the high parallel efficiency of these computations. For problems requiring non-uniform mesh refinement and enrichment, load unbalance is addressed by defining a larger number of small local problems than the number of parallel processors and by sorting and solving the local problems based on estimates of their workload. A simple and effective estimate of the largest number of processors where load balance among processors is maintained is also proposed. Several three-dimensional fracture mechanics problems aiming at investigating the accuracy and parallel performance of the proposed GFEM are analyzed.     

Parallel implementation of the finite element method using compressed data structures

This paper presents a parallel implementation of the finite element method designed for coarse-grain distributed memory architectures. The MPI standard is used for message passing and tests are run on a PC cluster and on an SGI Altix 350. Compressed data structures are employed to store the coefficient matrix and obtain iterative solutions, based on Krylov methods, in a subdomain-by-subdomain approach. Two mesh partitioning schemes are compared: non-overlapping and overlapping. The pros and cons of these partitioning methods are discussed. Numerical examples of symmetric and non-symmetric problems in two and three dimensions are presented.     

Tensor objects in finite element programming

This paper describes a novel programming tool, nDarray, which is designed using an Object Oriented Paradigm (OOP) and implemented in the C++ programming language. Finite element equations, represented in terms of multidimensional tensors are easily manipulated and programmed. The usual matrix form of the finite element equations are traditionally coded in FORTRAN, which makes it difficult to build and maintain complex program systems. Multidimensional data systems and their implementation details are seldom transparent and thus not easily dealt with and usually avoided. On the other hand, OOP together with efficient programming in C++ allows building new concrete data types, namely tensors of any order, thus hiding the lower level implementation details. These concrete data types prove to be quite useful in implementing complicated tensorial formulae associated with the numerical solution of various elastic and elastoplastic problems in solid mechanics. They permit implementing complex nonlinear continuum mechanics theories in an orderly manner. Ease of use and the immediacy of the nDarray programming tool in constitutive driver programming and in building finite element classes will be shown. © 1998 John Wiley & Sons, Ltd.     

A coupled meshfree/finite element method for automotive crashworthiness simulations

Mesh distortion induced numerical instability is a major roadblock in automotive crashworthiness finite element simulations. Remedies such as wrapping elements with null shells and deletion of distorted meshes have been adopted but none of them seems robust enough to survive various scenarios. Meshfree methods have been developed over the past almost twenty years in view of their capabilities in dealing with large material deformation and separation, but have remained in academic research due to their unaffordable high computational cost in solving large-scale industrial applications. This paper presents a coupled meshfree/finite-element method which allows engineers to model the severe deformation area with the meshfree method while keeping the remaining area modeled by the finite element methods. The method is implemented into LS-DYNA version 971 and its later versions so that it is available for automotive crashworthiness simulations. In the paper, one linear patch test and three crash examples are presented to demonstrate the accuracy of the meshfree formulation, its effectiveness in resolving mesh distortion difficulty, and the efficiency of the coupled meshfree/finite element solver in handling large-scale models.     

Parallel finite element computation of free-surface flows

In this paper we present parallel 2D and 3D finite element computation of unsteady, incompressible free-surface flows. The computations are based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation, which takes automatically into account the motion of the free surface. The free-surface height is governed by a kinematic free-surface condition, which is also solved with a stabilized formulation. The meshes consist of triangles in 2D and triangular-based prism elements in 3D. The mesh update is achieved with general or special-purpose mesh moving schemes. As examples, 2D flow past spillway of a dam and 3D flow past a surface-piercing circular cylinder are presented.     

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.     

Coupling of a non‐overlapping domain decomposition method for a nodal finite element method with a boundary element method

Non‐overlapping domain decomposition techniques are used to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the so‐called cross‐points, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting from the discretization process. Only small size finite element linear systems and a dense linear system related to a simple boundary integral equation are solved at each iteration and each of them can be solved in a stable way. We also show how to choose the parameter defining the augmented local matrices in order to improve the convergence. Several numerical simulations in 2D and 3D validating the treatment of the cross‐points and illustrating the strategy to accelerate the iterative procedure are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A multiscale finite element method for modeling fully coupled thermomechanical problems in solids

This article proposes a two‐scale formulation of fully coupled continuum thermomechanics using the finite element method at both scales. A monolithic approach is adopted in the solution of the momentum and energy equations. An efficient implementation of the resulting algorithm is derived that is suitable for multicore processing. The proposed method is applied with success to a strongly coupled problem involving shape‐memory alloys.Copyright © 2012 John Wiley & Sons, Ltd.     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

一种基于Layerwise理论的压电复合材料层合圆柱壳机电耦合有限单元

基于Reddy的Layerwise理论,对含压电铺层的复合材料层合壳的静力响应特性进行了理论研究。基于Layerwise理论,推导了含压电层的复合材料层合壳的应变分量与电场强度表达式。利用Hamilton原理和变分法,推导了压电智能层合壳的欧拉-拉格朗日方程,并采用有限元解法,建立了相应的有限元控制方程及其机电耦合刚度矩阵。通过算例结果与文献中的精确解和试验值进行了对比,表明相较于传统的经典层合板壳理论,本文理论方法的有效性和优势性;并分析了径厚比等参量对两端简支压电智能层合壳静力响应值的影响规律。      

A finite element method for elasto-plastic structures and contact problems by parametric quadratic programming

In this paper, the stiffness matrix of a contact element is introduced by means of a penalty function expression of the contact pressure and frictional force. The contact condition and the flow rule are expressed by the same form as in a non-associated plastic flow problem. A unified PQP (Parametric Quadratic Programming) model related to contact problems as well as to elasto-plastic structures is constructed. A series of PQP formulae for contact problems and elastic-plastic structures is derived in the text, and some numerical examples are illustrated as well.     

Two-dimensional analysis of anisotropic crack problems using coupled meshless and fractal finite element method

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with fractal the finite element method (FFEM) for analyzing homogeneous, anisotropic, and two dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element-free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T − stress. The numerical results based on all four orthotropic cases show that SIFs and T − stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions.