Die shape optimal design in three-dimensional shape metal extrusion by the finite element method

A new approach to die shape optimal design in shape extrusion is presented. In this approach, the design problem is formulated as an optimization problem incorporating the three-dimensional finite element analysis model, and optimization of the die shape is conducted on the basis of the design sensitivities. The approach is applied to the determination of the die shapes for extrusion of parts with various cross sections including polygons and T sections. © 1998 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.     

Assumed stress function finite element method: Two-dimensional elasticity

A complementary energy-based finite element formulation, using assumed stress functions as the approximating functions, is developed for the linear elastic two-dimensional stress analysis. It features the use of blending function interpolants, enabling the convenient representation of traction boundary conditions, which in the past have posed difficulties. A family of rectangular elements is constructed. Numerical results assessing the behaviour of these elements are presented. An advantage of this approach is in the accurate prediction of stress distributions.     

An expert system for the optimal mesh design in the hp-version of the finite element method

This paper suggests a simple expert system frame and provides the domain knowledge for the optimal mesh design and the prediction of the error in the energy norm for the problem of plane elasticity using the hp-extension in the finite element method. The expert system monitors the progress of the analysis, guides the user through the various steps and is able to reason about its own advice. In an example the user–expert communication is shown and the superiority of the results is demonstrated.     

A numerical method for the optimal blank shape design

This paper describes a numerical procedure for the blank shape design of thin metallic parts obtained by stamping. The objective is to determine the initial blank shape knowing the geometry of the desired 3D CAD part. The numerical procedure consists of two stages: At first, an estimation of the initial blank shape is given using the one step inverse approach (IA). Then, update of the blank shape is continued by iterations combining optimization algorithms and finite element analysis (FEA). The numerical procedure for the blank shape design is tested in the case of an industrial stamping process where the part is formed using a manual press without blank-holder. The proposed numerical procedure can provide very quickly the optimal blank shape in a few iterations.     

Application of the finite element method to an optimal control problem

The finite element method is applied to a typical problem in the optimal control theory. A finite element formulation is first set up, and then applied, as an example, to a problem of minimizing the transfer time of a low-thrust rocket vehicle between the orbits of Earth and Mars. A procedure which combines the gradient projection technique with the Newton-Raphson method is introduced to obtain numerically the solution of the minimum time transfer problem from the discretized functional. Numerical results show that the application of the finite element method to optimal control problems seems promising as well as encouraging.     

Dynamical micromagnetics by the finite element method

We developed a new numerical procedure to study dynamical behavior in micromagnetic systems. This procedure solves the damped Gilbert equation for a continuous magnetic medium, including all interactions in standard micromagnetic theory in three-dimensional regions of arbitrary geometry and physical properties. The magnetization is linearly interpolated in each tetrahedral element in a finite element mesh from its value on the nodes, and the Galerkin method is used to discretize the dynamic equation. We compute the demagnetizing field by solution of Poisson's equation and treat the external region by means of an asymptotic boundary condition. The procedure is implemented in the general purpose dynamical micromagnetic code (GDM). GDM uses a backward differential formula to solve the stiff ordinary differential equations system and the generalized minimum residual method with an incomplete Cholesky conjugate gradient preconditioner to solve the linear equations. GDM is fully parallelized using MPI and runs on massively parallel processor supercomputers, clusters of workstations, and single processor computers. We have successfully applied GDM to studies of the switching processes in isolated prolate ellipsoidal particles and in a system of multiple particles     

Elastic-plastic fracture analysis and design using the finite element method

The purpose of analysis in fracture mechanics is to determine characterising parameters which reflect the influence of loading and geometry on the crack tip environment of flawed bodies. Here practical methods are developed which permit the determination of such parameters in general situations. Extensive use of finite element methods has been made to provide relevant field values which are then manipulated to determine the required parameters. However the choice of method to determine field values is arbitrary and is dictated by the ease with which such field values may be found. It is in the manipulation of these values that the fracture mechanics philosophy is introduced.Contributions are made in three areas. First economic methods for the determination of the linear fracture mechanics parameter in general stiuations are developed which are of direct relevance to design procedures. Detailed discussion of the Dugdale model of fracture behaviour is then given and a general method for determining Dugdale model solutions is provided. This method is used to provide solutions for standard specimen geometries and it is suggested that such solutions will enable a rational evaluation of the general applicability of the model. However the method is such that, should sufficient confidence in the model be established, design calculations on its premises may be performed. Finally, it is demonstrated that materials which allow extensive plastic flow at a flaw tip prior to fracture may be analysed using the basic ideas of fracture mechanics. It is shown that a line integral provides a flaw tip environment parameter for materials deforming according to the physically appropriate Prandtl-Reuss laws of plasticity. It is hoped that these results will indicate a rational approach to correlating fracture behaviour in such situations.     

Two-dimensional finite element method for stress intensity factor using adaptive mesh strategy

Finite element analysis (FEA) combined with the concepts of Linear Elastic fracture mechanics (LEFM) provides a practical and convenient means to study the fracture and crack growth of materials. A numerical analysis (FEM) of cracks was developed to derive the SIF for two different geometries, i.e., a rectangular plate with half circle-hole and central edge crack plate in tension loading conditions. The onset criterion of crack propagation is based on the stress intensity factor, which is the most important parameter that must be accurately estimated and facilitated by the singular element. Displacement extrapolation technique (DET) is employed, to obtain the stress intensity factors (SIFs) at crack tip. The fracture is modeled by the splitting node approach and the trajectory follows the successive linear extensions of each crack increment. These comprehensive tests are evaluated and compared with other relevant numerical and analytical results obtained by other researchers.     

Analysis of acoustic resonator with shape deformation using finite element method

An acoustic resonator with shape deformation has been analysed using the finite element method. The shape deformation is such that the volume of the resonator remains constant. The effect of deformation on the resonant frequencies is studied. Deformation splits the degenerate frequencies.     

Magnetic resonance imaging gradient coil design by combiningoptimization techniques with the finite element method

In this paper, the optimization techniques of complex method, steepest descent, and conjugate gradient are investigated in terms of their convergence behaviors. The conjugate gradient method is then combined with finite element analysis techniques to develop a magnetic resonance imaging (MRI) G_z gradient coil design strategy which maximizes the field linearity within a specified region of interest. It is found that conjugate gradient optimization in conjunction with the finite element method is a powerful and flexible coil design approach with the potential to incorporate complex coil geometries, inhomogeneous media, and transient current excitation     

Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, K_I and K_II, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or K_I and K_II. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

Fractal finite element method based shape sensitivity analysis of multiple crack system

This paper presents fractal finite element based continuum shape sensitivity analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates.     

Application of the finite element method to the design of permanent magnets

A new method of determining the lengths of magnets in a magnetic circuit by using the finite element method has been developed. This method has the advantage that the lengths of magnets which produce the prescribed flux distribution can be directly calculated. In this paper, the error of this method is discussed at first, and then an example of application determining the shape of a magnet is shown. This method is effective for the design of magnetic circuits consisting of several permanent magnets and the determination of the shapes of magnets.     

A finite element approach to optimal design of plastic structures in plane stress

Plane stress structures of any shape and boundary conditions are simulated by finite element models with homogeneous stress and strain fields in each element. Besides the given live loads, dead loads (such as self-weight) depending on the unknown thickness distribution are allowed for. Possible practical requirements on geometry (minimum thickness, areas of equal thickness, areas of thickness variation in a prescribed way) are taken into account. Yield surfaces of the materials are piecewise linearized. On this basis the minimum weight design problem is formulated in terms of a linear program. This program is dualized and the pair of dual programming problems is discussed. The mechanical interpretation leads to the generalization of some known limit design theory results. Some numerical examples are given at the end.     

Solidification in castings by finite element method

Abstract

A self-contained CAD (computer aided design) system capable of analyzing foundry casting processes in sand and gravity dies is being developed at the University College of Swansea. The work involves preprocessing, postprocessing, and a finite element code with some novel numerical techniques. The solidification of castings is a heat transfer problem involving phase change, which may occur in a narrow range of temperatures. To simulate the phenomena accurately, very fine meshes must be used and the solution of such a system becomes very expensive. In the Swansea system, an adaptive remeshing technique is introduced, which tracks the moving front of the phase change zone. At every time step, a scan is made to determine the points at which phase change is occurring, so that the remeshing may be done to produce a refined mesh at such points. The computing process is then continued. Examples have illustrated that the method is efficient and accurate. In addition, an interfacial heat transfer model is introduced to improve the simulation of the casting process. Advective heat transfer in the liquid is also modelled.

MST/1041     

Study of rectangular waveguides with grooves of irregular shape using the finite element method

Electromagnetic wave propagation through grooved waveguides is studied using the finite element method (FEM). The effect of grooves of irregular shape on TE₁₀, TE₂₀ mode frequencies and passband is studied. The variation in cutoff frequencies for TE₁₀, TE₂₀ mode and passband is observed.     

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.     

State-space representation and optimal control of non-linear material deformation using the finite element method

A state-space model for representing the non-linear material deformation and an optimal control scheme for obtaining desired process conditions in the deforming material are presented in this paper. The formulation is general for various metal-forming processes including forging and extrusion operations. The state variables selected in the formulation are the die/billet contact nodal velocities and the nodal velocities of the critical finite elements of the billet. The control input is the ram velocity, which is determined by using the linear quadratic regulator (LQR) theory to maintain desired strain rates within the selected finite elements. The influence of an optimally designed ram velocity on the deforming material is studied using performance measures. This paper includes the development of the state-space model from non-linear finite element formulation, optimal control strategy and numerical example cases with discussions.