A minimal element distortion strategy for computational mesh dynamics

Mesh motion strategy is one of the key points in many fluid–structure interaction (FSI) problems. Due to the increasing application of FSI to solve the current challenging engineering problems, this topic has become of great interest. There are several different strategies to solve this problem, some of them use a discrete and lumped spring–mass system to propagate the boundary motion into the volume mesh, and many others use an elastostatic problem to deform the mesh. In all these strategies there is always risk of producing an invalid mesh, i.e. a mesh with some elements inverted. Normally this condition is irreversible and once an invalid mesh is obtained it is difficult to continue. In this paper the mesh motion strategy is defined as an optimization problem. By its definition this strategy can be classified as a particular case of an elastostatic problem where the material constitutive law is defined in terms of the minimization of certain energy functional that takes into account the degree of element distortion. Some advantages of this strategy are its natural tendency to high quality meshes, its robustness and its straightforward extension to 3D problems. Several examples included in this paper show these capabilities. Even though this strategy seems to be very robust it is not able to recover a valid mesh starting from an invalid one. This improvement is left for future work. Copyright © 2006 John Wiley & Sons, Ltd.     

3-D shape optimal design and automatic finite element regridding

A unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed. A domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed. For numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface. In shape design problems, nodal points of the finite element model move as the shape changes. A method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid. For numerical implementation of the continuum shape design sensitivity analysis and automatic regridding, an established finite element analysis code is used. To demonstrate the feasibility of the method developed, shape design optimization of a main engine bearing cap is carried out as an example.     

A class of data structures for 2-D and 3-D adaptive mesh refinement

A ‘family’ of tree data structures for adaptive mesh refinement is described and details concerning the associated logic are provided. The data structures encompass triangular elements and quadrilateral elements in two dimensions and quadrilateral bricks in three dimensions. Furthermore, both linear (bilinear) and quadratic (biquadratic) element types, respectively, are developed. Representative refinement results are given for the bilinear, trilinear and biquadratic types and associated performance studies made for the refinement procedure.     

Finite element simulation and modeling of 2-D arrays for 3-D ultrasonic imaging

Issues of modeling and design of 2-D arrays in three dimensions with finite element code are discussed. These ultrasonic arrays are used for real time dynamic imaging of the heart. Topics include optimization, sensitivity, and performance and methods to speed up the run times required for computer simulations of large three-dimensional models. Empirical results from 45×45 2-D arrays are also presented     

Automatic 3-D crack propagation calculations: a pure hexahedral element approach versus a combined element approach

This article presents an evaluation of two different crack prediction approaches based on a comparison of the stress intensity factor distribution for three example problems. A single edge notch specimen and a quarter circular corner crack specimen subjected to shear displacements and a three point bend specimen with a crack inclined to the mid-plane are examined. The stress intensity factors are determined from the singular stress field close to the crack front. Two different fracture criteria are adopted for the calculation of an equivalent stress intensity factor and crack deflection angle. The stress intensity factor distributions for both numerical methods agree well to available reference solutions. Deviations are recorded at crack front locations near the free surface probably due to global contraction effects and the twisting behaviour of the crack front. Crack propagation calculations for the three point bending specimen give results that satisfy intuitive expectations. The outcome of the study encourages further pursuit of a crack propagation tool based on a combination of elements.     

2-D multiplane finite element technique for solving a class of 3-D problems

A finite element technique, for efficient solution of a class of 3-D elasticity problems, is presented. In this method, standard 2-D finite elements are used along with a ‘connector’ element. An element, previously used to model material interfaces, is shown to provide the properties for use as a ‘connector’ element, if input variables are redefined. The accuracy of the technique is illustrated with a sample solution.     

Improvements in the reliability and element quality of parallel tetrahedral mesh generation

The paper presents a parallel tetrahedral mesh generation approach based on recursive bidivisions using triangular surfaces. Research was conducted for addressing issues concerning mesh generation reliability and element quality. A novel procedure employing local modification techniques is proposed for repairing the intersecting interdomain mesh instead of directly repeating the bidivision procedure, which improves the robustness of the complete meshing procedure significantly. In addition, a new parallel quality improvement scheme is suggested for optimizing the distributed volume meshes. The scheme is free of any communication cost and highly efficient. Finally, mesh experiments of hundreds of millions of elements are performed to demonstrate the reliability, effectiveness and efficiency of the proposed method and its potential applications to large‐scale simulations of complex aerodynamics models. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element analysis of 3-D eddy currents

The authors review formulations of three-dimensional (3-D) eddy current problems in terms of various magnetic and electric potentials. The differential equations and boundary conditions are formulated to include the necessary gauging conditions and thus to ensure the uniqueness of the potentials. Different sets of potentials can be used in distinct subregions, thus facilitating an economic treatment of various types of problems. A novel technique for interfacing conducting regions with an electric vector and a magnetic scalar potential to eddy-current-free regions with a magnetic vector potential is described. Finite-element solutions to several large eddy-current problems are presented     

Automatic construction of 3-D models in multiple scale analysis

This paper discusses techniques for the automatic construction of numerical analysis models for multiple scale analyses which employ interacting models at two, or more, physical scales. Consideration is given to the methods to define the geometric representations and generate the discretizations needed by the numerical analysis procedures. The application of the techniques to multichip modules and composite structures, with interacting macromechanical and micromechanical level analyses, is demonstrated. In the multichip module analyses both heat conduction and thermomechanical analysis are performed using different numerical analysis techniques, and the two interaction of the analyses at the through levels is through a basic global/local methodology. The composite structure analysis considers crack propagation at the micromechanical level interacting with the macromechanical analysis through finite element based adaptive multiscale analysis. In both example applications the focus of the discussion is on the automatic construction of the required geometric models and their automatic discretization.     

3-D boundary element method for cracks in a plate

New fundamental solutions which automatically satisfy boundary conditions at the interfaces of an elastic plate perfectly bonded to two elastic halfspaces are implemented in a 3-D boundary element method (BEM) for crack problems. The BEM features a new integration scheme for highly singular kernels. The capability is achieved through a part analytic and part numerical integration procedure, such that the analytic part of the integration is similar for all slip/opening variations, ‘Part-through’ elliptic cracks in an elastic plate with traction-free surfaces are analysed and the stress intensity factor (SIF) values along the crack front are found to compare favourably with widely accepted numerically obtained SIF results by Raju and Newman.¹     

Finite element computation of 3-D nonlinear magnetic field inclaw-pole electric machine using two scalar potentials, hybrid elementsand automatic mesh generation

The basic equations of two scalar potentials and their finite-element discretized formulations are discussed. In order to compute the 3-D magnetic field in a complex solution domain, a method of hybrid elements consisting of tetrahedron, triangular prism, and arbitrary hexahedron is used, and its feasibility is justified by theoretical analysis and numerical examples. This method is successfully used to calculate the 3-D nonlinear magnetic field in a claw-pole electric machine with a power rating of 500 W     

Numerical modeling of ductile crack growth in 3-D using computational cell elements

This study describes a 3-D computational framework to model stable extension of a macroscopic crack under mode I conditions in ductile metals. The Gurson-Tvergaard dilatant plasticity model for voided materials describes the degradation of material stress capacity. Fixed-size, computational cell elements defined over a thin layer at the crack plane provide an explicit length scale for the continuum damage process. Outside this layer, the material remains undamaged by void growth, consistent with metallurgical observations. An element vanish procedure removes highly voided cells from further consideration in the analysis, thereby creating new tractionfree surfaces which extend the macroscopic crack. The key micro-mechanics parameters are D, the thickness of the computational cell layer, and f ₀ , the initial cell porosity. Calibration of these parameters proceeds through analyses of ductile tearing to match R-curves obtained from testing of deep-notch, through-crack bend specimens. The resulting computational model, coupled with refined 3-D meshes, enables the detailed study of non-uniform growth along the crack front and predictions of specimen size, geometry and loading mode effects on tearing resistance, here described by J-a curves. Computational and experimental studies are described for shallow and deep-notch SE(B) specimens having side grooves and for a conventional C(T) specimen without side grooves. The computational models prove capable of predicting the measured R-curves, post-test measured crack profiles, and measured load-displacement records.     

2-D Wavelet Segmentation in 3-D T-Ray Tomography

In this letter, segmentation techniques for terahertz (T-ray) computed tomographic (CT) imaging are investigated. A set of linear image fusion and novel wavelet scale correlation segmentation techniques is adopted to achieve material discrimination within a 3-D object. The methods are applied to a T-ray CT image dataset taken from a plastic vial containing a plastic tube. This setup simulates the imaging of a simple nested organic structure, which provides an indication of the potential for using T-ray CT imaging to achieve T-ray pulsed signal classification of heterogeneous layers     

Boundary element analysis of 3-D magnetostatic problems usingscalar potentials

A boundary element formulation for 3-D nonlinear magnetostatic field problems using the total scalar potential and its normal derivative as unknowns is described. The boundary integral equation is derived from a differential equation for the total scalar potential where a nonlinear operator term can be separated from a linear term. The nonlinear term leads to a volume integral which can be treated as a known forcing function within an iterative solution process. An additional forcing term results from the magnetic excitation coil system. It is shown that the line integral of the magnetic source field which can be defined outside of the current-carrying regions as a gradient of a scalar potential acts as an excitation term. The proposed method is applied to a test problem where an iron cube immersed in the magnetic field of a cylindrical coil is investigated. The numerical results for different saturation stages are compared with finite element method (FEM) calculations. The comparison with FEM calculations shows a good agreement only in highly saturated iron parts     

3-D schapery representation for non-linear viscoelasticity and finite element implementation

On the basis of the one-dimensional Schapery representation for non-linear viscoelasticity, a three-dimensional constitutive model incorporating the effects of temperature and physical ageing is developed for isotropic non-linear viscoelastic materials. Adopting the assumption that the hydrostatic and deviatoric responses are uncoupled, the contitutive equation is expressed in incremental form for both compressible and incompressible materials, with the hereditary integral updated at the end of each time increment by recursive computation. The proposed model is implemented in the finite element package MARC. Numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms.Laboratory for Engineering Mechanics, Delft University of Technology, P. O. Box 5033, 2600 GA Delft, The Netherlands     

A 3-D boundary element formulation of viscoelastic media with gravity

This paper presents a boundary element formulation for 3-D linear and viscoelastic bodies subjected to the body force of gravity. The Laplace transformation is first used to suppress the time variable, and solutions of displacements and stresses are found in the transformed domain. The time domain solutions are then found by an accurate and efficient numerical inversion method which requires only real calculations for all quantities. Input and output data, and solutions in the transformed and time domains are connected through an Interactive Data Language code written by the authors. While particular solutions of stresses and displacements related to the body force of gravity (which is applied at time t = 0 and is kept constant) are derived, the Green's functions in the Laplace domain are obtained through the correspondence principle. The new formulation has been implemented into an existing 3-D BEM program, and several numerical examples involving 3-D viscoelastic bodies are presented. Although the discussion in this paper focuses on Maxwell viscoelastic and isotropic media, other linear isotropic and even anisotropic viscoelastic models can also be incorporated, without difficulty, into the 3-D viscoelastic BEM program.     

Green's matrices for the 2-D Lame system in regions of irregular shape

A method is proposed for the construction of Green's matrices for mixed boundary value problems in regions of irregular shape for the displacement formulation of the plane problem in theory of elasticity. The method is based on the boundary integral equation approach where a kernel matrix B satisfies the 2-D homogeneous Lame system inside the region. This leads to a regular boundary integral equation where the compensating load is applied to the boundary. The Green's matrix is consequently expressed in terms of the kernel matrix B, the fundamental solution matrix of the homogeneous Lame system and a kernel matrix of the inverse regular integral operator. To calculate the stress components, the kernel matrices are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stresses.