Die shape design optimization of sheet metal stamping process using meshfree method

A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity‐based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum‐based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation‐dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring‐back effect and to show the feasibility of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.     

On variational sensitivity analysis and configurational mechanics

This contribution is concerned with the application of variational design sensitivity analysis in the context of structural optimization and configurational mechanics. In both disciplines we consider variations of the material configuration and we use techniques from variational sensitivity analysis in order to solve these problems. We derive the physical and material residual problem in one step by using standard optimization procedures. Furthermore, we investigate the sensitivity of the physical as well as the material residual problem and obtain the coupled saddle point problem based on these sensitivities. Both problems are coupled by the pseudo load operator, which plays an important role by the solution of structural optimization problems. By means of computational examples from mesh optimization and shape optimization, we demonstrate the capability of the proposed theoretical framework.     

Meshless shape design sensitivity analysis and optimization for contact problem with friction

 In this paper, a continuum-based shape design sensitivity formulation for a frictional contact problem with a rigid body is proposed using a meshless method. The contact condition is imposed using the penalty method that regularizes the solution of variational inequality. The shape dependency of the contact variational form with respect to the design velocity field is obtained. The dependency of the response with respect to the shape of the rigid body is also considered. It is shown that the sensitivity equation needs to be solved at the final converged load step for the frictionless contact problem, whereas for the frictional contact case the sensitivity solution is needed at the converged configuration of each load step because the sensitivity of the current load step depends on that of the previous load step. The continuum-based contact formulation and consistent linearization is critical for accurate shape design sensitivity results. The accuracy of the proposed method is compared with the finite difference result and excellent agreement is obtained for a door seal contact example. A design optimization problem is formulated and solved to reduce the contact gap opening successfully in a demonstration of the proposed method.     

Hybrid stress finite element model for non-linear shell problems

The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.     

The inner structure of sensitivities in nodal based shape optimisation

The pseudo load matrix and the sensitivity matrix dominate design sensitivity analysis of shape optimisation problems. They describe how a structure reacts on an imposed design modification. We analyse these matrices for the model problem of nodal based shape optimisation by a singular value decomposition and show that they contain additional valuable information which is not yet used either in theory or computation of shape optimisation. The inner structure of the sensitivities is capable to formulate reduced quadratic sub-problems within the sequential quadratic programming approach. We also tackle the problem of indefinite Hessian matrices in nodal based shape optimisation. Furthermore, we avoid jagged boundaries and obtain mesh-independent optimised structures applying density filtering technique to shape optimisation. Overall, we emphasise an enhanced analysis of sensitivities and point to unused substantial capabilities.     

Non-singular symmetric boundary element formulation for elastodynamics

In this paper, a non-singular boundary element formulation for 3D-elastostatics and 3D-elastodynamics is presented. The proposed method is based on a generalized variational principle. A weighted superposition of static fundamental solutions is used for the field approximation in the domain, whereas the displacement and stress field on the boundary are interpolated by well-known polynomial shape functions. By separating time- and space-dependence a symmetric equation of motion is derived with time-independent mass and stiffness matrix. The domain integral over inertia terms, leading to the mass matrix, is analytically transformed to the boundary. Thus, a boundary only formulation is derived. Comparing numerical results with analytical solutions clearly shows that the obtained system of equations is well-suited for dynamic problems.     

Configuration design sensitivity analysis and optimization of beam structures

 A general method for configuration design sensitivity analysis over a three-dimensional beam structure is developed based on a variational formulation of the classical beam in linear elasticity. A sensitivity formula is derived based on a variational equation for the beam structure using the material derivative concept and adjoint variable method. The formulation considers not only the shape variation in a three dimensional direction, which includes translational as well as rotational change of the beam but also the orientation angle variation of the beam's cross section. The sensitivity formula can be evaluated with generality and ease even by employing a piecewise linear design velocity field despite the fact that the bending model is a fourth order differential equation. The design sensitivity analysis is implemented using the post-processing data of a commercial code ANSYS. Several numerical examples are given to show the excellent accuracy of the method. Optimization is carried out for a tilted arch bridge and an archgrid structure to show the method's applicability. Received 29 September 2001 / Accepted 20 March 2002     

Implementation of an ANCF beam finite element for dynamic response optimization of elastic manipulators

Theoretical and practical aspects of an absolute nodal coordinate formulation (ANCF) beam finite element implementation are considered in the context of dynamic transient response optimization of elastic manipulators. The proposed implementation is based on the introduction of new nodal degrees of freedom, which is achieved by an adequate nonlinear mapping between the original and new degrees of freedom. This approach preserves the mechanical properties of the ANCF beam, but converts it into a conventional finite element so that its nodal degrees of freedom are initially always equal to zero and never depend explicitly on the design variables. Consequently, the sensitivity analysis formulas can be derived in the usual manner, except that the introduced nonlinear mapping has to be taken into account. Moreover, the adjusted element can also be incorporated into general finite element analysis and optimization software in the conventional way. The introduced design variables are related to the cross-section of the beam, to the shape of the (possibly) skeletal structure of the manipulator and to the drive functions. The layered cross-section approach and the design element technique are utilized to parameterize the shape of individual elements and the whole structure. A family of implicit time integration methods is adopted for the response and sensitivity analysis. Based on this assumption, the corresponding sensitivity formulas are derived. Two numerical examples illustrate the performance of the proposed element implementation.     

Shape sensitivity analysis of magnetic forces

This paper presents a shape sensitivity analysis of magnetic forces evaluated using the Maxwell stress tensor and the finite element method. The formulation is based upon a discrete approach which takes the analytical derivatives of the finite element equations with respect to the shape variables and also on the adjoint variable method in order to carry out the derivation procedure. Sensitivity analysis is developed in the context of the axisymmetrical nonlinear magnetostatic field problem with a modified magnetic vector potential as state variable. Numerical results are presented to validate this methodology. Shape sensitivity analysis is then applied to the optimization of the force-displacement characteristic of a linear actuator. A sequential quadratic programming method is used in the optimization process     

A modified extended finite element method approach for design sensitivity analysis

This paper describes a modified extended finite element method (XFEM) approach, which is designed to ease the challenge of an analytical design sensitivity analysis in the framework of structural optimisation. This novel formulation, furthermore labelled YFEM, combines the well‐known XFEM enhancement functions with a local sub‐meshing strategy using standard finite elements. It deviates slightly from the XFEM path only at one significant point but thus allows to use already derived residual vectors as well as stiffness and pseudo load matrices to assemble the desired information on cut elements without tedious and error‐prone re‐work of already performed derivations and implementations. The strategy is applied to sensitivity analysis of interface problems combining areas with different linear elastic material properties. Copyright © 2015 John Wiley & Sons, Ltd.     

On the equivalent static load method for flexible multibody systems described with a nonlinear finite element formalism

The equivalent static load (ESL) method is a powerful approach to solve dynamic response structural optimization problems. The method transforms the dynamic response optimization into a static response optimization under multiple load cases. The ESL cases are defined based on the transient analysis response whereupon all the standard techniques of static response optimization can be used. In the last decade, the ESL method has been applied to perform the structural optimization of flexible components of mechanical systems modeled as multibody systems (MBS). The ESL evaluation strongly depends on the adopted formulation to describe the MBS and has been initially derived based on a floating frame of reference formulation. In this paper, we propose a method to derive the ESL adapted to a nonlinear finite element approach based on a Lie group formalism for two main reasons. Firstly, the finite element approach is completely general to analyze complex MBS and is suitable to perform more advanced optimization problems like topology optimization. Secondly, the selected Lie group formalism leads to a formulation of the equations of motion in the local frame, which turns out to be a strong practical advantage for the ESL evaluation. Examples are provided to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite rotation exact geometry solid-shell element for laminated composite structures through extended SaS formulation and 3D analytical integration

A nonlinear exact geometry hybrid-mixed four-node solid-shell element using the sampling surfaces (SaS) formulation is developed for the analysis of the second Piola-Kirchhoff stress that extends the authors' finite element (Int J Numer Methods Eng. 2019;117:498-522) to laminated composite shells. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as basic shell unknowns. The external surfaces and interfaces are also included into a set of SaS. The proposed hybrid-mixed solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated by efficient three-dimensional (3D) analytical integration. As a result, the developed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements. It could be useful for the 3D stress analysis of thick and thin doubly curved laminated composite shells because the SaS formulation gives the possibility to obtain the 3D solutions with a prescribed accuracy.     

声场-结构耦合系统灵敏度分析及优化设计研究

给出了低频声－结构耦合系统的有限元方程，并在此基础上提出声一结构耦合系统的包含尺寸和形状设计变量的优化设计模型，建立基于灵敏度分析求解的优化设计方法，重点推导了耦合系统的特征频率和声压级响应关于设计变量的灵敏度方程。在JIFEX软件中实现上述理论和算法，并通过灵敏度比较和优化设计的数值算例，进一步说明该研究方法对声结构耦合系统的工程设计具有实用意义。     

Mixed finite element method for coupled thermo‐hydro‐mechanical process in poro‐elasto‐plastic media at large strains

A mixed finite element for coupled thermo‐hydro‐mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three‐filed variational principle. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto‐plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct differentiation method for sensitivity analysis based on transfer matrix method for multibody systems

On one hand, the new version of transfer matrix method for multibody systems (NV‐MSTMM), has been proposed by formulating transfer equations of elements in acceleration level instead of position level as in the original discrete time transfer matrix method of multibody systems to study multibody system dynamics. This new formulation avoids local linearization and allows using any integration algorithms. On the other hand, sensitivity analysis is an important way to improve the optimization efficiency of multibody system dynamics. In this paper, a totally novel direct differentiation method based on NV‐MSTMM for sensitivity analysis of multibody systems is developed. Based on direct differentiation method, sensitivity analysis matrix for each kind of element is established. By assembling transfer matrices and sensitivity analysis matrices based on differentiation law of multiplication, the sensitivity analysis equation of overall transfer equation is deduced. The computing procedure of the proposed method is also presented. All these improvements as well as three numerical examples show that the direct differentiation method based on NV‐MSTMM is suitable for optimizing the dynamic sensitivity in multi–rigid‐body systems.     

SPATIAL STABILITY ANALYSIS OF THIN-WALLED SPACE FRAMES

A clearly consistent finite element formulation for spatial stability analysis of thin-walled space frames is presented by applying linearized virtual work principle and introducing Vlasov's assumption. The improved displacement field for unsymmetric thin-walled cross-sections is introduced based on inclusion of second-order terms of finite rotations, and the potential energy corresponding to the semitangential moments is consistently derived. In the present formulation, displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis, and all bending-torsional coupling effects due to unsymmetric cross-sections are taken into account. For finite element analysis, cubic Hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of Hermitian space frame element. Also, load correction stiffness matrices for off-axis point loadings are derived based on the second-order rotation terms. Finite element solutions for the spatial buckling analysis of thin-walled space frames are compared with available solutions and other researcher's results.     

Pseudo-excitation-method-based sensitivity analysis and optimization for vehicle ride comfort

In this article, sensitivity analysis formulae for optimizing vehicle suspension systems are derived using the pseudo excitation method (PEM). A spatial finite element model is used to describe the dynamic behaviour of a vehicle running on a randomly uneven road, of which the irregularity is assumed to be a Gaussian random process. Based on the random equations of motion with the right-hand side random acceleration replaced by a pseudo acceleration excitation, various first and second orders of sensitivity formulae are calculated conveniently by differentiating these equations. The optimal solutions when vehicle ride comfort is the objective function are derived by means of these flexibilities. The optimization efficiency and the computational accuracy are numerically justified.     

A new boundary-type finite element for 2-D- and 3-D-elastic structures

In this paper we discuss the theoretical and numerical formulation of 3-D Trefitz elements. Starting from the variational principle with the so-called hybrid stress method, the trial functions for the stresses have to fulfil the Beltrami equations, which means also the compatibility equations for the strains. The divergence theorem can be applied, and one arrives at a pure boundary formulation in the sense of the Trefftz method. Besides the resulting variational formulation, different regularizations of the interelement conditions are investigated by numerical tests. Two examples show the numerical efficiency of the derived elements. First, a geometric linear 3-D example is presented to show the effects on distorted element meshes. The third example shows the geometrically non-linear analysis of a shallow cylindrical shell segment under a singe load.     

Variational inequality‐based framework of discontinuous deformation analysis

For modeling discrete particle‐block systems, a new framework of discontinuous deformation analysis is established on the basis of finite‐dimensional variational inequality. The presented method takes into account the contacts, the rolling resistance, and the tensile resistance of cemented interface among particles and blocks using the corresponding variational or quasivariational inequalities. The new formulation avoids using the artificial springs that are usually indispensable in many conventional methods dealing with similar discrete problems and conveniently integrates the rigid circle particles, the nonrigid ring particles, and the arbitrary shape blocks into a uniform framework. The proposed discontinuous deformation analysis approach is further coupled with the finite element method using a node‐based composite contact matrix and several simple transformation matrices to solve practical problems. A particle/block‐based composite contact matrix is constructed to further broaden the application of the proposed method. The accuracy, robustness, and capability of the presented method are demonstrated with examples.     

Efficient handling of stability problems in shell optimization by asymmetric ‘worst‐case’ shape imperfection

The paper presents an approach to shape optimization of proportionally loaded elastic shell structures under stability constraints. To reduce the stability‐related problems, a special technique is utilized, by which the response analysis is always terminated before the first critical point is reached. In this way, the optimization is always related to a precritical structural state. The necessary load‐carrying capability of the optimal structure is assured by extending the usual formulation of the optimization problem by a constraint on an estimated critical load factor. Since limit points are easier to handle, the possible presence of bifurcation points is avoided by introducing imperfection parameters. They are related to an asymmetric shape perturbation of the structure. During the optimization, the imperfection parameters are updated to get automatically the ‘worst‐case’ pattern and amplitude of the imperfection. Both, the imperfection parameters and the design variables are related to the structural shape via the design element technique. A gradient‐based optimizer is employed to solve the optimization problem. Three examples illustrate the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.