Adaptive generation of hexahedral element meshes for finite element analysis of metal plastic forming process

A method for the adaptive generation of hexahedral element mesh based on the geometric features of solid model is proposed. The first step is to construct the refinement information fields of source points and the corresponding ones of elements according to the surface curvature of the analyzed solid model. A thickness refinement criterion is then used to construct the thickness-based refinement information field of elements from digital topology. The second step is to generate a core mesh through removing all the undesired elements using even and odd parity rules. Then the core mesh is magnified in an inside–out manner method through a surface node projection process using the closest position approach. Finally, in order to match the mesh to the characteristic boundary of the solid model, a threading method is proposed and applied. The present method was applied in the mesh construction of different engineering problems. The resulting meshes are well-shaped and capture all the geometric features of the original solid models.     

基于质感要素的产品材质意象设计研究

目的为了探讨产品材质意象设计的有效方法,研究利用质感要素来构建材质意象评价体系。方法利用色彩理论、主成分分析法等方法,确定产品色彩、材料、表面处理工艺的质感要素、质感要素评价等级和质感要素评价系数。利用质感要素定量描述确定材质、质感要素、材质意象的数学表达式,并构建三者之间的关系模型。结果以水杯为例,利用最小二乘法确定水杯材质意象评价模型。经验证,水杯材质意象评价模型的准确率为80%。结论质感要素评价等级、质感要素评价系数为材质意象设计提供了参数化方法。产品材质意象关系模型较好地反映了材质、质感要素、材质意象之间的内在关系。利用提出的材质意象设计方法,可计算任意产品的材质意象值,为产品材质意象设计提供理论指导。     

Expert systems and finite element structural analysis — a review

Finite element analysis of many engineering systems is practised more as an art than as a science. It involves high level expertise (analytical as well as heuristic) regarding problem modelling (e.g. problem specification, choosing the appropriate type of elements etc.), optical mesh design for achieving the specified accuracy (e.g. initial mesh selection, adaptive mesh refinement), selection of the appropriate type of analysis and solution routines and, finally, diagnosis of the finite element solutions. Very often such expertise is highly dispersed and is not available at a single place with a single expert. The design of an expert system, such that the necessary expertise is available to a novice to perform the same job even in the absence of trained experts, becomes an attractive proposition. In this paper, the areas of finite element structural analysis which require experience and decision-making capabilities are explored. A simple expert system, with a feasible knowledge base for problem modelling, optimal mesh design, type of analysis and solution routines, and diagnosis, is outlined. Several efforts in these directions, reported in the open literature, are also reviewed in this paper.     

An adaptive mesh refinement technique for the analysis of shear bands in plane strain compression of a thermoviscoplastic solid

We have developed an adaptive mesh refinement technique that generates elements such that the integral of the second invariant of the deviatoric strain-rate tensor over an element is nearly the same for all elements in the mesh. It is shown that the finite element meshes so generated are effective in resolving shear bands, which are narrow regions of intense plastic deformation that form in high strain-rate deformation of thermally softening viscoplastic materials. Here we assume that the body is deformed in plane strain compression at a nominal strain-rate of 5000 sec^-1, and model a material defect by introducing a temperature perturbation at the center of the block.     

Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test

We present a geometrically non‐linear assumed strain method that allows for the presence of arbitrary, intra‐finite element discontinuities in the deformation map. Special attention is placed on the coarse‐mesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit. Given an underlying mesh and an arbitrary failure surface, we first construct an enriched approximation for the deformation map with the non‐linear analogue of the extended finite element method (X‐FEM). With regard to the richer space of functions spanned by the gradient of the enriched approximation, we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain. In particular, in those elements intersected by the failure surface, we construct enhanced strain approximations which are orthogonal to piecewise‐constant stress fields. Contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields. Importantly, the present formulation gives rise to a symmetric tangent stiffness matrix, even in localized elements. The present modification also allows for the satisfaction of a discontinuous patch test, wherein two different constant stress fields (on each side of the failure surface) lie in the solution space. We demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface, particularly for nearly incompressible material response. Additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive mesh refinement technique for two-dimensional shear band problems

We have developed an adaptive mesh refinement technique that rezones the given domain for a fixed number of quadrilateral elements such that fine elements are generated within the severely deformed region and coarse elements elsewhere. Loosely speaking, the area of an element is inversely proportional to the value of the deformation measure at its centroid. Here we use the temperature rise at a material point to gauge its deformations which is reasonable for the shear band problem since the material within the shear band is deformed intensely and is heated up significantly. It is shown that the proposed mesh refinement technique is independent of the initial starting mesh, and that the use of an adaptively refined mesh gives thinner shear bands, and shaper temperature rise and the growth of the second invariant of the plastic strain-rate within the band as compared to that for a fixed mesh having the same number of nodes. The method works well even when the deformation localizes into more than one narrow region.     

FINITE ELEMENT REMESHING: A METAL FORMING APPROACH FOR QUADRILATERAL MESH GENERATION AND REFINEMENT

The paper presents an automatic finite element remeshing system for quadrilateral elements consisting of modules for mesh generation, densification, smoothing and interpolation of field variables. The mesh generator takes into account the contour of the old mesh, eventual interference with dies and the plastic deformation of the material. An initial coarse mesh is created by utilizing a grid-based approach for creating well-shaped internal elements, in conjunction with a nodal connection approach based on constrained Delaunay triangulation, for linking with the boundary. Subsequent local mesh refinement is performed according to parameters depending on past, present and predicted future deformation related field variables; being, respectively, the strain gradient and strain rate distribution in relation with the velocity field, element size and quality. Smoothing is accomplished using an iterative Laplacian repositioning method. As illustrated in the presented examples this overall strategy ensures a robust and efficient remeshing scheme for finite element simulation of bulk metal-forming processes. © 1997 by John Wiley & Sons, Ltd.     

Distortion,degeneracy and rezoning in finite elements — A survey

The results obtained from finite element analysis are significantly affected by the quality of elements. In certain applications like shape optimization, crash analysis, metal forming, fluid flow analysis, and large displacement analysis, the finite element mesh is systematically updated in an iterative process. In such situations, in spite of an ideal starting mesh, the quality of elements could deteriorate, causing severly distorted elements. In extreme cases, the elements become degenerate and further progress of analysis is restricted. An understanding of the methods of quantifying element distortion helps in identifying ‘bad’ geometry and in deciding when to remesh. Knowledge about geometric configurations which cause degeneracy assists in controlling degeneracy during the analysis. This paper contains a survey of available distortion measures and degeneracy conditions for various elements in two and three dimensions. It is a review of the literature in this field in the last two decades. A brief review of rezoning is also included, since it is one of the more popularly used methods to correct a distorted mesh.     

3D mesh refinement in compliance with a specified node spacing function

A simple and efficient refinement procedure for the three-dimensional tetrahedral element mesh based on successive bisection of edges is proposed. The quality of the elements generated can be guaranteed if the subdivision is performed in the sequence according to the length of the line segments to be divided. Such an order of priority can be determined by a simple sorting process on all the line segments for which refinement is needed. This list of ordered line segments has to be updated from time to time to take into account of the new line segments generated during the subdivision process. From the examples studied, the CPU time for mesh refinement bears a linear relationship with the number of elements generated, with a refinement rate of more than 50 000 elements per second on a IBM Power Station 3BT. Shape optimization procedures can be applied to the refined mesh to further improve the quality of the elements. The refinement scheme is useful as part of a general three-dimensional mesh generation package, or as the mesh refinement module in an adaptive finite element analysis.     

Efficient wave propagation simulation on quadtree meshes using SBFEM with reduced modal basis

We apply a combination of the transient scaled boundary finite element method (SBFEM) and quadtree‐based discretization to model dynamic problems at high frequencies. We demonstrate that the current formulation of the SBFEM for dynamics tends to require more degrees of freedom than a corresponding spectral element discretization when dealing with smooth problems on regular domains. Thus, we improve the efficiency of the SBFEM by proposing a novel approach to reduce the number of auxiliary variables for transient analyses. Based on this improved SBFEM, we present a modified meshing procedure, which creates a quadtree mesh purely based on the geometry and allows arbitrary sizes and orders of elements, as well as an arbitrary number of different materials. The discretization of each subdomain is created automatically based on material parameters and the highest frequency of interest. The transition between regions of different properties is straightforward when using the SBFEM. The proposed approach is applied to image‐based analysis with a particular focus on geological models. Copyright © 2016 John Wiley & Sons, Ltd.     

Microstructure-based simulation of thermomechanical behavior of composite materials by object-oriented finite element analysis

While it is well recognized that microstructure controls the physical and mechanical properties of a material, the complexity of the microstructure often makes it difficult to simulate by analytical or numerical techniques. In this paper we present a relatively new approach to incorporate microstructures into finite element modeling using an object-oriented finite element technique. This technique combines microstructural data in the form of experimental or simulated microstructures, with fundamental material data (such as elastic modulus or coefficient of thermal expansion of the constituent phases) as a basis for understanding material behavior. The object-oriented technique is a radical departure from conventional finite element analysis, where a “unit-cell” model is used as the basis for predicting material behavior. Instead, the starting point of object-oriented finite element analysis is the actual microstructure of the material being investigated. In this paper, an introduction to the object-oriented finite element approach to microstructure-based modeling is provided with two examples: SiC particle-reinforced Al matrix composites and double-cemented WC particle-reinforced Co matrix composites. It will be shown that object-oriented finite element analysis is a unique tool that can be used to predict elastic and thermal constants of the composites, as well as salient effects of the microstructure on local stress state.     

Synthesis of shape and topology of multi-material structures with a phase-field method

In this paper, we present a phase-field method to the problem of shape and topology synthesis of structures with three materials. A single phase model is developed based on the classical phase-transition theory in the fields of mechanics and material sciences. The multi-material synthesis is formulated as a continuous optimization problem within a fixed reference domain. As a single parameter, the phase-field model represents regions made of any of the three distinct material phases and the interface between the regions. The Van der Waals–Cahn-Hilliard theory is applied to define a dynamic process of phase transition. The Γ-convergence theory is used for an approximate numerical solution to this free-discontinuity problem without any explicit tracking of the interface. Within this variational framework, we show that the phase-transition theory leads to a well-posed problem formulation with the effects of “domain regularization” and “region segmentation” incorporated naturally. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. It is further suggested that such a phase-field approach may represent a promising alternative to the widely-used homogenization models for the design of heterogeneous materials and solids, with a possible extension to a general model of multiple material phases.     

Dedicated finite elements for electrode thin films on quartz resonators

The accuracy of the finite element analysis for thickness shear quartz resonators is a function of the mesh resolution; the finer the mesh resolution, the more accurate the finite element solution. A certain minimum number of elements are required in each direction for the solution to converge. This places a high demand on memory for computation, and often the available memory is insufficient. Typically the thickness of the electrode films is very small compared with the thickness of the resonator itself; as a result, electrode elements have very poor aspect ratios, and this is detrimental to the accuracy of the result. In this paper, we propose special methods to model the electrodes at the crystal interface of an AT cut crystal. This reduces the overall problem size and eliminates electrode elements having poor aspect ratios. First, experimental data are presented to demonstrate the effects of electrode film boundary conditions on the frequency-temperature curves of an AT cut plate. Finite element analysis is performed on a mesh representing the resonator, and the results are compared for testing the accuracy of the analysis itself and thus validating the results of analysis. Approximations such as lumping and Guyan reduction are then used to model the electrode thin films at the electrode interface and their results are studied. In addition, a new approximation called merging is proposed to model electrodes at the electrode interface.     

A ‘FE-Meshfree’ TRIA3 element based on partition of unity for linear and geometry nonlinear analyses

A new 3-node triangular element is developed on the basis of partition of unity (PU) concept. The formulation employs the parametric shape functions of classical triangular element (TRIA3) to construct the PU and the least square point interpolation method to construct the local displacement approximation. The proposed element synergizes the individual merits of finite element method and meshfree method. Moreover, the usual linear dependence problem associated with PU finite elements is eliminated in the present element. Application of the element to several linear and geometric nonlinear problems shows that the proposed element gives a performance better than that of classical linear triangular as well as linear quadrilateral elements, and comparable to that of quadratic quadrilateral element. The proposed element does not necessitate a new mesh or additional nodes in the mesh. It uses the same mesh as the classical TRIA3 element and is able to give more accurate solution than the TRIA3 element.     

Converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for FEM accuracy and efficiency

This paper presents a computational method for converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for improved solution accuracy and computational efficiency of finite element analysis. The proposed method performs this conversion by inserting layers of prism elements and deleting tetrahedral elements in sweepable sub‐domains, in which cross‐sections remain topologically identical and geometrically similar along a certain sweeping path. The total number of finite elements is reduced because roughly three tetrahedral elements are converted to one prism element. The solution accuracy of the finite element analysis improves since a prism element yields a more accurate solution than a tetrahedral element due to the presence of higher‐order terms in the shape function. Only previously known method for creating such a prism–tetrahedral hybrid mesh was to manually decompose a target volume into sweepable and non‐sweepable sub‐volumes and mesh each of the sub‐volumes separately. Unlike the previous method, the proposed method starts from a cross‐section of a tetrahedral mesh and replaces the tetrahedral elements with layers of prism elements until prescribed quality criteria can no longer be satisfied. A series of computational fluid dynamics simulations and structural analyses have been conducted, and the results verified a better performance of prism–tetrahedral hybrid mesh. Copyright © 2009 John Wiley & Sons, Ltd.     

A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling

We propose a multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. Our framework combines two or more competing objective functions into a single objective function to be solved using one of various multiobjective optimization methods. Methods within our framework are able to optimize various aspects of the mesh such as the element shape, element size, associated PDE interpolation error, and number of inverted elements, but the improvement is not limited to these categories. The strength of our multiobjective mesh optimization framework lies in its ability to be extended to simultaneously optimize any aspects of the mesh and to optimize meshes with different element types. We propose the exponential sum, objective product, and equal sum multiobjective mesh optimization methods within our framework; these methods do not require articulation of preferences. However, the solutions obtained satisfy a sufficient condition of weak Pareto optimality. Experimental results show that our multiobjective mesh optimization methods are able to simultaneously optimize two or more aspects of the mesh and also are able to improve mesh qualities while eliminating inverted elements. We successfully apply our methods to real‐world applications such as hydrocephalus treatment and shape optimization. Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling of micro-crack growth during thermal shock based on microstructural images of thermal barrier coatings

Based on digital image processing theory and finite element mesh generation principle, a methodology is proposed to model the micro-crack growth of thermal barrier coatings (TBCs) during thermal shock with the aid of finite element program. Firstly, a microstructural image of plasma sprayed TBCs is transferred to digital image; secondly, a finite element grid model is generated by thresholding segmentation according to the actual microstructure; finally, based on the finite element grid model, the Tuler–Butcher failure criterion is employed to model the micro-crack growth of TBCs during thermal shock. The numerical simulation result agrees well with the experimental result, and the methodology presented in this paper is found to be effective to model the micro-crack growth.     

Quality improvement of non‐manifold hexahedral meshes for critical feature determination of microstructure materials

This paper describes a novel approach to improve the quality of non‐manifold hexahedral meshes with feature preservation for microstructure materials. In earlier works, we developed an octree‐based isocontouring method to construct unstructured hexahedral meshes for domains with multiple materials by introducing the notion of material change edge to identify the interface between two or more materials. However, quality improvement of non‐manifold hexahedral meshes is still a challenge. In the present algorithm, all the vertices are categorized into seven groups, and then a comprehensive method based on pillowing, geometric flow and optimization techniques is developed for mesh quality improvement. The shrink set in the modified pillowing technique is defined automatically as the boundary of each material region with the exception of local non‐manifolds. In the relaxation‐based smoothing process, non‐manifold points are identified and fixed. Planar boundary curves and interior spatial curves are distinguished, and then regularized using B‐spline interpolation and resampling. Grain boundary surface patches and interior vertices are improved as well. Finally, the optimization method eliminates negative Jacobians of all the vertices. We have applied our algorithms to two beta titanium data sets, and the constructed meshes are validated via a statistics study. Finite element analysis of the 92‐grain titanium is carried out based on the improved mesh, and compared with the direct voxel‐to‐element technique. Copyright © 2009 John Wiley & Sons, Ltd.     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

Mesh motion techniques for the ALE formulation in 3D large deformation problems

Efficient mesh motion techniques are a key issue to achieve satisfactory results in the arbitrary Lagrangian–Eulerian (ALE) finite element formulation when simulating large deformation problems such as metal‐forming. In the updated Lagrangian (UL) formulation, mesh and material movement are attached and an excessive mesh distortion usually appears. By uncoupling mesh movement from material movement, the ALE formulation can relocate the mesh to avoid distortion. To facilitate the calculation process, the ALE operator is split into two steps at each analysis time step: UL step (where deformation due to loading is calculated without convective terms) and Eulerian step (where mesh motion is applied). In this work, mesh motion is performed by new nodal relocation methods, developed for eight‐node hexahedral elements, which can move internal and boundary nodes, improving and concentrating the mesh in critical zones. After mesh motion, data is transferred from the UL mesh to the relocated mesh using an expansion of stresses in a Taylor's series. Two numerical applications are presented, comparing results of UL and ALE formulation with results found in the literature. Copyright © 2004 John Wiley & Sons, Ltd.