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.     

Characterizing load transfer efficiency in double-walled carbon nanotubes using multiscale finite element modeling

Load transfer efficiency from matrix to carbon nanotubes (CNTs) plays an important role in the mechanical response of CNTs nanocomposites as it may affect the effectiveness of the nano-reinforcements. For double-walled carbon nanotubes (DWCNTs), the outer graphene layer as well as the inner layer may be responsible for the load bearing capacity. In this study, the load transfer efficiency within DWCNTs was investigated using a multiscale simulation scheme. The multiscale simulation consists of two steps. First, the atomistic behaviors between the adjacent graphite layers in DWCNTs were characterized using molecular dynamic (MD) simulation, from which a cylindrical equivalent continuum solid of DWCNTs with embedded spring elements was proposed to describe the interactions of neighboring graphene layers. Two kinds of interatomistic properties in DWCNTs, i.e., van der Walls (vdW) interactions and artificial build-up covalent bonds, were considered in the equivalent solid. Subsequently, the equivalent solid was implemented as reinforcement in the micromechanical model of CNTs nanocomposites for evaluating the load transfer efficiency. Results indicated that the DWCNTs with covalent bonds exhibit superior load transfer efficiency than those with only vdW interactions. In addition, when the DWCNTs get long, the load transfer efficiency of DWCNTs increases accordingly.     

Use of material forces in adaptive finite element methods

The theory of material forces for a hyperelastic material is briefly presented using the translational invariance of the control volume. The theory is derived for the dynamical setting, while the numerical implementation is limited to the static case. The finite element (FE) method is used to solve the standard field equations. After obtaining the solution the material force balance is consistently discretized with FE. As a result of this post-processing discrete material forces are obtained. They are then used to set up an adaptive scheme, in which the magnitude of the material forces acts as an indicator for mesh refinement. Special consideration is given to the boundary, where two different refinement strategies are proposed. The two strategies are compared by studying the refinement process for three examples.     

A computational library for multiscale modeling of material failure

We present an open-source software framework called PERMIX for multiscale modeling and simulation of fracture in solids. The framework is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH. Fracture on the continuum level is modeled by the extended finite element method (XFEM). Using several novel or state of the art methods, the piece software handles semi-concurrent multiscale methods as well as concurrent multiscale methods for fracture, coupling two continuum domains or atomistic domains to continuum domains, respectively. The efficiency of our open-source software is shown through several simulations including a 3D crack modeling in clay nanocomposites, a semi-concurrent FE-FE coupling, a 3D Arlequin multiscale example and an MD-XFEM coupling for dynamic crack propagation.     

基于耦合扩展多尺度有限元方法的功能梯度材料热应力分析

以高效模拟功能梯度材料(FGM)微观非均质性对整体热力学性能的影响为研究目的,通过随机形态描述函数(RMDF)法和体积分数的指数分布建立FGM二维微结构,在此基础上,发展了FGM热应力分析的耦合扩展多尺度有限元方法(CEMsFEM)。该方法基于扩展多尺度有限元方法(EMsFEM)的基本思想,对温度场和位移场构造数值基函数,以把微观非均质材料性质带到宏观响应中。同时为了考虑泊松效应导致的不同方向间的耦合作用,在位移场数值基函数中增加了耦合附加项。通过数值基函数建立宏微观单元信息的映射关系,在宏观尺度求解有效方程,节约计算量。为了更好地考虑微观载荷的影响,把结构的真实响应分解为宏观响应和微观扰动,进一步推导出修正的宏观载荷向量。通过不同体积分数分布的FGM在不同载荷工况下的热应力分析算例验证了本文中方法的正确性和有效性,最后讨论了微结构的尺寸效应对结构热力学响应的影响。     

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.     

Effect of orthotropic material on finite element modeling of completely dentate mandible

The objective of this investigation is to construct a high quality complete dentate mandible model with detailed biological structures, and assign mandibular bone with inherent orthotropic material characteristics. Three different types of scan data are used to elaborate detailed mandibular structures, including the cortical and cancellous bone, tooth enamel, dentin, periodontal ligament, temporal fossa, TMJ articular disk, temporal cartilage, and condylar cartilage. In addition, an extended orthotropic material assignment methodology based on harmonic fields is used to handle the alveolar ridge region of dentate mandible, to generate compatible orthotropic axes fields. The influence of orthotropic material on the biomechanical behavior of complete dentate mandible is analyzed compared with commonly used isotropic model. The result revealed that the orthotropic model would induce higher stress values and more well-distributed stress pattern than the isotropic model, especially for the cancellous bone. And the orthotropic model would induce lower volumetric strain values than the isotropic model on the cortical bone. It was concluded that elastic orthotropy had a significant effect on the simulated stress value and distribution pattern, as well as the volumetric strain, and demonstrated the mechanical optimality of the mandible.     

Incomplete factorization for finite element methods

An incomplete factorization iterative scheme with ORTHOMIN acceleration is evaluated for three-dimensional finite element applications. Both heat conduction and combined heat and fluid flow are investigated. Computational efficiency is investigated for both linear (8 node) and quadratic (27 node) elements as well as for various degrees of LU Decomposition. The factorization schemes worked well with both types of elements, and a variable factorization scheme, which distributes the level of factorization, performed well on difficult heat and fluid flow problems.     

A generalized phase field multiscale finite element method for brittle fracture

A generalized multiscale finite element method is introduced to address the computationally taxing problem of elastic fracture across scales. Crack propagation is accounted for at the microscale utilizing phase field theory. Both the displacement-based equilibrium equations and phase field state equations at the microscale are mapped on a coarser scale. The latter is defined by a set of multinode coarse elements, where solution of the governing equations is performed. Mapping is achieved by employing a set of numerically derived multiscale shape functions. A set of representative benchmark tests is used to verify the proposed procedure and assess its performance in terms of accuracy and efficiency compared with the standard phase field finite element implementation.     

Laser generation of Lamb waves for defect detection: experimental methods and finite element modeling

The propagation of Lamb waves generated by a pulsed laser beam in an aluminum sheet is modeled using finite element analysis, and the interaction with defects is studied and compared to experimental results. The ultrasonic Lamb waves are detected by an electromagnetic acoustic transducer (EMAT). The frequency content of the received wave is shown to be enhanced when the generation point is situated directly over the defect in both the modeled and experimental cases. Time-frequency analysis using a Wigner transform has enabled individual modes to be identified.     

Seat-cushion and soft-tissue material modeling and a finite element investigation of the seating comfort for passenger-vehicle occupants

Improved seating comfort is an important factor that most car manufacturers use to distinguish their products from those of their competitors. In today’s automotive engineering practice, however, design and development of new, more comfortable car seats is based almost entirely on empiricism, legacy knowledge and extensive, time-consuming and costly prototyping and experimental/field testing. To help accelerate and economize the design/development process of more-comfortable car seats, more extensive use of various computer aided engineering (CAE) tools will be necessary. However, before the CAE tools can be used more successfully by car-seat manufacturers, issues associated with the availability of realistic computer models for the seated human, the seat and the seated-human/seat interactions as well as with the establishment of objective seating-comfort quantifying parameters must be resolved.In the present work, detailed finite element models of a prototypical car seat and of a seated human are developed and used in the investigation of seated-human/seat interactions and the resulting seating comfort. To obtain a fairly realistic model for the human, a moderately detailed skeletal model containing 16 bone assemblies and 15 joints has been combined with an equally detailed “skin” model of the human. The intersection between the two models was then used to define the muscular portion of the human. Special attention in the present work has been given to realistically representing/modeling the materials present in different sections of the car seat and the seated human. The models developed in the present work are validated by comparing the computational results related to the pressure distribution over the seated-human/seat interface with their open-literature counterparts obtained in experimental studies involving human subjects.     

Computational modeling of mixed-mode fatigue crack growth using extended finite element methods

The extended finite element method (XFEM) combined with a cyclic cohesive zone model (CCZM) is discussed and implemented for analysis of fatigue crack propagation under mixed-mode loading conditions. Fatigue damage in elastic-plastic materials is described by a damage evolution equation in the cohesive zone model. Both the computational implementation and the CCZM are investigated based on the modified boundary layer formulation under mixed-mode loading conditions. Computational results confirm that the maximum principal stress criterion gives accurate predictions of crack direction in comparison with known experiments. Further popular multi-axial fatigue criteria are compared and discussed. Computations show that the Findley criterion agrees with tensile stress dominant failure and deviates from experiments for shear failure. Furthermore, the crack propagation rate under mixed mode loading has been investigated systematically. It is confirmed that the CCZM can agree with experiments.     

Image-based finite element mesh construction for material microstructures

One way of computing the macroscopic behavior of a material sample with complex microstructure is to construct a finite element model based on a micrograph of a representative slice of the material. The quality of the results produced with such a model obviously depends on the quality of the constructed mesh. In this article, we describe a set of routines that modify and improve the quality of a 2D mesh. Most of the routines are guided by an effective element “energy” functional, which takes into account the shape quality of the elements and the homogeneity of the elements as determined from an underlying segmented image. The interfaces and boundaries in the image arise naturally from the segmentation process. From these routines, we construct a close-to-automatic mesh generator that requires only a few inputs, such as the linear sizes of the largest and smallest features in the micrograph.     

Natural hierarchical refinement for finite element methods

Current formulations of adaptive finite element mesh refinement seem simple enough, but their implementations prove to be a formidable task. We offer an alternative point of departure which yields equivalent adapted approximation spaces wherever the traditional mesh refinement is applicable, but our method proves to be significantly simpler to implement. At the same time it is much more powerful in that it is general (no special tricks are required for different types of finite elements), and applicable for some newer approximations where traditional mesh refinement concepts are not of much help, for instance on subdivision surfaces. Copyright © 2003 John Wiley & Sons, Ltd.     

Galerkin finite element methods for wave problems

We compare here the accuracy, stability and wave propagation properties of a few Galerkin methods. The basic Galerkin methods with piecewise linear basis functions (called G1FEM here) and quadratic basis functions (called G2FEM) have been compared with the streamwise-upwind Petrov Galerkin (SUPG) method for their ability to solve wave problems. It is shown here that when the piecewise linear basis functions are replaced by quadratic polynomials, the stencils become much larger (involving five overlapping elements), with only a very small increase in spectral accuracy. It is also shown that all the three Galerkin methods have restricted ranges of wave numbers and circular frequencies over which the numerical dispersion relation matches with the physical dispersion relation — a central requirement for wave problems. The model one-dimensional convection equation is solved with a very fine uniform grid to show the above properties. With the help of discontinuous initial condition, we also investigate the Gibbs’ phenomenon for these methods.     

Time finite element methods for structural dynamics

Time finite element methods are developed for the equations of structural dynamics. The approach employs the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Results are presented from finite difference analyses of the time-discontinuous Galerkin and least-squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics. These results show that, for particular interpolations, the time finite element method exhibits improved accuracy and stability.     

On material forces and finite element discretizations

 The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations. Received 12 December 2001 / Accepted 18 March 2002     

Surrogate modeling of multiscale models using kernel methods

This work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods. The key element is to learn the interface between the different scales using a fast surrogate for the microscale model, which is given by multivariate kernel expansions. The expansions are computed using statistically representative samples of input and output of the microscale model. We apply both support vector machines and a vectorial kernel greedy algorithm as learning methods. We demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines. We consider, first, a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and, second, a model for simulation of saturation overshoots in porous media involving nonclassical shock waves. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple scale finite element methods

New temporal and spatial discretization methods are developed for multiple scale structural dynamic problems. The concept of fast and slow time scales is introduced for the temporal discretization. The required time step is shown to be dependent only on the slow time scale, and therefore, large time steps can be used for high frequency problems. To satisfy the spatial counterpart of the requirement on time step constraint, finite-spectral elements and finite wave elements are developed. Finite-spectral element methods combine the usual finite elements with the fast convergent spectral functions to obtain a faster convergence rate; whereas, finite wave elements are developed in parallel to the temporal shifting technique. Therefore, the spatial resolution is increased substantially. These methods are especially applicable to structural acoustics and linear space structures. Numerical examples are presented to illustrate the effectiveness of these methods.