A formal theory for estimating defeaturing-induced engineering analysis errors

Defeaturing is a popular CAD/CAE simplification technique that suppresses ‘small or irrelevant features’ within a CAD model to speed-up downstream processes such as finite element analysis. Unfortunately, defeaturing inevitably leads to analysis errors that are not easily quantifiable within the current theoretical framework.In this paper, we provide a rigorous theory for swiftly computing such defeaturing-induced engineering analysis errors. In particular, we focus on problems where the features being suppressed are cutouts of arbitrary shape and size within the body. The proposed theory exploits the adjoint formulation of boundary value problems to arrive at strict bounds on defeaturing induced analysis errors. The theory is illustrated through numerical examples.     

Estimating defeaturing-induced engineering analysis errors for arbitrary 3D features

Defeaturing is a widely used technique in preparing CAD models for meshing for use in finite element analysis. However, defeaturing invariably leads to analysis errors that affect the final analysis results. A practical numerical approach is proposed in this paper to estimate this defeaturing-induced engineering analysis error, in terms of changes to particular quantities of engineering interest. The proposed approach can handle arbitrary 3D negative and positive features of any size. They can either be free or be subject to external loading. Our estimator is expressed in a residual form defined over a suppressed feature, and can be explicitly evaluated using solutions of the defeatured model or its extensions. Experimental results on practical engineering parts show the performance of the approach.     

Automatic surface repairing,defeaturing and meshing algorithms based on an extended B-rep

This paper presents an extended surface boundary representation (B-rep), where each topology entity can have dual geometric representations to accommodate various defects (e.g., gaps and overlaps) commonly present in CAD models. Keeping a uniform B-rep and the unsuppressed geometry data enables the use of various existing repairing, defeaturing and meshing algorithms to process CAD models with small gaps and overlaps on surface boundaries. The continuous geometry of the input model remains untouched in the repairing, defeaturing and meshing process, and the output mesh is loyal to this geometry. Such feature is often desirable in numerical simulations that require meshes with high geometry fidelity.     

Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and the point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a “fixed basis function” finite element shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite element matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C⁰ problem; the extension to three-dimensional problems is straight-forward. Received December 5, 1999?Revised mansucript received July 6, 2000     

Optimization of hyperelastic materials with isotropic damage

In many practical problems, engineering structures under repeated loading exhibit softening material behaviour. The complex micromechanical processes yielding the observed loss of stiffness are often described phenomenologically on the macroscopic level by damage mechanics. A finite strain elastic constitutive model incorporating an isotropic damage mechanism was developed by Simo (1987). The additional theoretical and computational enhancements for utilizing this damage model and the associated finite element formulation for optimization purposes are outlined in this paper.?The structural response and its sensitivity expressions at a given time and position depend on the response and the response sensitivities of all previous locations and times. The expressions for variational design sensitivity analysis within damage mechanics are fully stated and related to prior work on history dependent material behaviour such as Prandtl-Reuss elastoplasticity, see Barthold and Wiechmann (1997) and Wiechmann et al. (1997). The theoretical details and the corresponding finite element formulation were described in the paper by Firuziaan (1998).?New problem functions based on the internal variables are shown to be adequate for controlling and optimizing the damage process.?Numerical experiments illustrate the method proposed and the efficiency of the overall optimization procedure. Received April 29, 1999     

Numerical comparison between two cost functions in contact shape optimization

A finite element approach to shape optimization in a 2D frictionless contact problem for two different cost functions is presented in this work. The goal is to find an appropriate shape for the contact boundary, performing an almost constant contact-stress distribution. The whole formulation, including the mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by a direct modification of the geometry throughB-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data. Augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization) are used to obtain numerical results.     

Feature-based modeling for automatic mesh generation

Automatic meshing algorithms for finite element analysis are based on a computer understanding of the geometry of the part to be discretized. Current mesh generators understand the part as either a boundary representation, an octree, or a point set. A higher-level understanding of the part can be achieved by associating engineering significance and engineering data, such as loading and boundary conditions, with generic shapes in the part. This technique, called feature-based modeling, is a popular approach to integrating computer-aided design (CAD) and computer-aided manufacturing through the use of machinable shapes in the CAD model. It would seem that feature-based design also could aid in the finite element mesh generation process by making engineering information explicit in the model.This paper describes an approach to feature-based mesh generation. The feature representation of a fully functioning feature-based system that does automatic process planning and inspection was extended to include finite element mesh generation. This approach is based on a single feature representation that can be used for design, finite element analysis, process planning, and inspection of prismatic parts. The paper describes several advantages that features provide to the meshing process, such as improved point sets and a convenient method of simplifying the geometry of the model. Also discussed are possible extensions to features to enhance the finite element meshing process.     

A nonlinear visco-elastic constitutive model for large rotation finite element formulations

Although all known materials have internal damping that leads to energy dissipation, most existing large deformation visco-elastic finite element formulations are based on linear constitutive models or on nonlinear constitutive models that can be used in the framework of an incremental co-rotational finite element solution procedure. In this investigation, a new nonlinear objective visco-elastic constitutive model that can be implemented in non-incremental large rotation and large deformation finite element formulations is developed. This new model is based on developing a simple linear relationship between the damping forces and the rates of deformation vector gradients. The deformation vector gradients can be defined using the decomposition of the matrix of position vector gradients. In this paper, the decomposition associated with the use of the tangent frame that is equivalent to the QR decomposition is employed to define the matrix of deformation gradients that enter into the formulation of the viso-elastic constitutive model developed in this investigation. Using the relationship between the deformation gradients and the components of the Green–Lagrange strain tensor, it is shown that the damping forces depend nonlinearly on the strains and linearly on the classical strain rates. The relationship between the damping forces and strains and their rates is used to develop a new visco-elastic model that satisfies the objectivity requirements and leads to zero strain rates under an arbitrary rigid body displacement. The linear visco-elastic Kelvin–Voigt model frequently used in the literature can be obtained as a special case of the proposed nonlinear model when only two visco-elastic coefficients are used. As demonstrated in this paper, the use of two visco-elastic coefficients only leads to viscous coupling between the deformation gradients. The model developed in this investigation can be used in the framework of large deformation and large rotation non-incremental solution procedure without the need for using existing co-rotational finite element formulations. The finite element absolute nodal coordinate formulation (ANCF) that allows for straightforward implementation of general constitutive material models is used in the validation of the proposed visco-elastic model. A comparison with the linear visco-elastic model is also made in this study. The results obtained in this investigation show that there is a good agreement between the solutions obtained using the proposed nonlinear model and the linear model in the case of small deformations.     

Nth-order stiffness sensitivities in structural analysis

This paper presents a general expression for theNth-order stiffness sensitivities in linear elastic frames. It is based on modelling the structure as being composed of unimodal elements. It is shown that the sensitivity of the structural response to the variation of the stiffness of an arbitrary component depends only on the corresponding elemental displacements. These are the nodal displacements due to nodal element loads applied to the structure at the end nodes of the considered element. Therefore, on the basis of one structural analysis we obtain the sensitivity of the structure to the variation of a given stiffness, to any order and for all loading conditions. Partial derivatives with respect to several element stiffnesses are obtained from the elemental displacements of the considered elements. The method is equally applicable to more general finite element models. It requires, however, the preliminary decomposition of the finite elements into their unimodal components.     

Finite element analysis of the Girkmann problem using the modern hp-version and the classical h-version

We perform finite element analysis of the so called Girkmann problem in structural mechanics. The problem involves an axially symmetric spherical shell stiffened with a foot ring and is approached (1) by using the axisymmetric formulation of linear elasticity theory and (2) by using a dimensionally reduced shell-ring model. In the first approach the problem is solved with a fully automatic hp-adaptive finite element solver whereas the classical h-version of the finite element method is used in the second approach. We study the convergence behaviour of the different numerical models and show that accurate stress resultants can be obtained with both models by using effective post-processing formulas.     

Shape sensitivity analysis using low- and high-order finite elements

A general procedure to perform the sensitivity analysis for the shape optimal design of elastic structures is proposed. The method is based on the implicit differentiation of the discretized equilibrium equations used in the finite element method (FEM). The so-called semianalytical approach is followed, that is, finite differences are used to differentiate the finite element matrices. The technique takes advantage of the geometric modeling concepts typical of the computer-aided design (CAD) technology used in the creation of a compact design model. This procedure is largely independent of the types of finite elements used in the analysis and has been implemented in ah-version andp-version finite element program. Very accurate and stable shape sensitivity derivatives were obtained from both programs over a wide range of finite difference step sizes. It is shown that the method is computationally efficient, general, and relatively easy to implement. Some classical shape optimal design problems have been solved using the CONLIN optimizer supplied with these gradients.     

SWL550型十字轴万向联轴器结构建模及有限元分析

利用现代设计技术对SWL550型十字轴万向联轴器进行结构设计及强度计算。通过对十字轴万向联轴器的力学分析,得到重要零部件的载荷边界条件。基于CATIA软件建立重要零部件模型,通过CATIA的有限元分析模块对其进行有限元分析,危险截面的Von Mises应力与理论计算值基本吻合。用数字化模型代替传统的实物联轴器实验,简化了联轴器的设计开发过程。     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

Generating contours on FEM/BEM higher-order surfaces using Java 3D textures

An efficient technique to visualize primary and secondary results for combined finite element method/boundary element method models as contours is presented. The technique is based on dividing higher-order surfaces into triangles and on using texture interpolation to produce contour plots. Since results of high accuracy with significant gradients can be obtained using sparse meshes of boundary elements and finite elements, special attention is devoted to element face subdivision. Subdivision density is defined on the basis of both face edge curvature and ranges of result fields over element faces. Java 3D API is employed for code development.     

Toward a Meta-Model for Computational Engineering

Geometric modeling and finite element analysis have matured in recent decades. Both methods are used extensively in engineering design. However, the link between geometric modeling, physical modeling and finite element analysis is currently cumbersome, error-prone, and ad-hoc. Topological domain modeling provides the missing link. In this paper, we propose a combined topological modeling and finite element modeling method that allows not only topological modeling, but also promotes geometric and physical modeling, by providing a topological base space for the definition of finite element meshes, fields, and the definition and solution of boundary value problems. We call the method the Constructive Topological Domain Method (CTDM). In this method, Primitive Topological Domains (PTDs), each possessing a natural coordinate space, are combined in multiple n-dimensional Cartesian coordinate spaces, called charts, using generalizations of Boolean set operations, to create Constructed Topological Domains (CTDs) capable of acting as the base spaces of fiber bundles. The charts are glued together to create an atlas, within which the CTD is defined. The fiber of the bundle may describe, in addition to geometry, physical fields like density, stress, and temperature. Finite element meshes may be defined upon each of the PTDs from which the CTD is constructed, enabling the definition and solution of boundary value problems, thus avoiding the difficult and messy problem of creating a single finite element mesh to represent the entire CTD. A modified finite element method, to handle the individually meshed PTDs, is described. The boundary conditions may be specified as analytical or as finite element-based fields upon each of the PTDs. The CTDM appears to be a promising approach to robust mathematical and computational modeling of physical objects. Simple examples are presented. ID="A1"Correspondance and offprint requests to: W. Gerstle, Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: gerstle@unm.edu     

A numerical study of singularly perturbed generalized Burgers-Huxley equation using three-step Taylor-Galerkin method

In the present work, a numerical study has been carried out for the singularly perturbed generalized Burgers-Huxley equation using a three-step Taylor-Galerkin finite element method. A Burgers-Huxley equation represents the traveling wave phenomena. In singular perturbed problems, a very small positive parameter, ?, called the singular perturbation parameter is multiplied with the highest order derivative term. As this parameter tends towards zero, the problem exhibits boundary layers. The traditional methods fail to capture the boundary layers when ? becomes very small. In this paper a three-step Taylor-Galerkin finite element method is used to capture the boundary layers. The method is third-order accurate and has inbuilt upwinding. Stability analysis has been carried out and the numerical results show that the method is efficient in capturing the boundary layers.     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.     

Efficient surrogate method for predicting pavement response to various tire configurations

A computationally efficient surrogate model was developed based on artificial neural networks (ANN) to investigate the effect of the new generation of wide-base tires on pavement responses. Non-uniform tire contact stress measurements were obtained using a stress-in-motion instrument. The measured 3-D contact stresses were applied on two extreme 3-D flexible pavement finite element models representing low-volume (thin) and high-volume (thick) roads. Eleven critical pavement responses were modeled at two different material properties input levels—detailed and simplified—depending on data availability. The results rendered by the ANN surrogate models were highly accurate with average prediction error less than 5 % and R-square values higher than 0.95. In addition, two sensitivity analyses were performed to investigate the variables effect on pavement responses. It was found that the type of tire (wide-base vs. dual tire assembly) is more influential than the inflation pressure on pavement responses. However, the tire inflation pressure seemed to have a significant effect on near-surface responses. The developed models were incorporated into a tool to assist designers and engineers in investigating the effect of the pavement responses of wide-base versus dual tire assembly under typical loading conditions and pavement structures.

     

针对白车身轻量化系数的结构灵敏度分析与软件开发

为实现白车身轻量化,以白车身零件厚度为优化变量,建立参数化模型。定义白车身静态扭转刚度工况并进行有限元分析,得到扭转刚度响应和轻量化系数,采用解析法推导轻量化系数对厚度的灵敏度。基于HyperMesh二次开发完成灵敏度分析流程自动化,求解白车身轻量化系数的灵敏度。根据灵敏度排序对白车身零件厚度进行优化,实现轻量化系数降低,扭转刚度提高,白车身质量减轻。