Configuration Design Sensitivity Analysis Research Articles

Isogeometric configuration design sensitivity analysis of geometrically exact shear-deformable beam structures

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method.

Isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using Jaumann strain formulation

Using an isogeometric approach, a continuum-based configuration design sensitivity analysis (DSA) method is developed for curved Kirchhoff beams with multi-patch junctions. Under the total Lagrangian formulation, large deformations considering the initial curvature of curved beams are described by geometrically exact beam theory (GEBT) and Jaumann strain formulation. In the isogeometric approach, the higher order continuity and the exact description of initial geometry are naturally embedded using NURBS basis functions. In multi-patch models, C0-continuity of physical displacement or C1-continuity of displacement component at junction is weakly imposed using the Lagrange multiplier method. The superior accuracy of isogeometric analysis (IGA) is verified through the comparison with the results of finite element analysis (FEA) using cubic Hermite interpolation. In the DSA, a material derivative is utilized and the kinematical description of GEBT is consistently employed to express orientation design variations. Contrary to the IGA-based DSA, the Hermite basis function explicitly depends on design in the FEA-based DSA due to its element length parameter. Moreover, since the design velocity field is approximated using the nodal velocity imposed at nodal tangential vector, the amount of design perturbations should be very small to obtain precise design sensitivity.

Isogeometric configuration design optimization of heat conduction problems using boundary integral equation

The shape variation of a domain naturally results in both shape and orientation variations, so called configuration variation, when employing a boundary integral equation (BIE) method. A configuration design sensitivity analysis (DSA) method is developed for steady state heat conduction problems using the boundary integral equations in an isogeometric approach, where NURBS basis functions in a CAD system are directly utilized in the response analysis. Thus, we can accomplish a seamless incorporation of exact geometry and the higher continuity into a computational framework. To enhance the accuracy of configuration design sensitivity, the CAD-based higher-order geometric information such as normal and tangent vectors is exactly embedded in the design sensitivity expressions. The necessary velocity field for configuration design obtained from the NURBS is analytically decomposed into shape and orientation velocity fields. It is shown to be essential to consider orientation variations and significant for accurate configuration sensitivity through comparison with finite differencing conventional BIE method. The developed isogeometric configuration DSA method turns out to be accurate compared with the analytic solution and the conventional DSA method. During the optimization, a mesh regularization scheme is employed to avoid excessive mesh distortion, which comes from significant design changes.

Design sensitivity analysis and optimization for minimizing springback of sheet‐formed part

AbstractThe springback is a manufacturing defect in the stamping process and causes difficulty in product assembly. An impediment to the use of lighter‐weight, higher‐strength materials in manufacturing is relative lack of understanding about how these materials respond to complex forming processes. The springback can be reduced by using an optimized combination of die, punch, and blank holder shapes together with friction and blank‐holding force. An optimized process can be determined using a gradient‐based optimization to minimize the springback. For an effective optimization of the stamping process, development of an efficient design sensitivity analysis (DSA) for the springback with respect to these process parameters is crucial. A continuum‐based shape and configuration DSA method for the stamping process has been developed using a non‐linear shell model. The material derivative is used to develop the continuum‐based design sensitivity. The design sensitivity equation is solved without iteration at each converged load step in the finite deformation elastoplastic non‐linear analysis with frictional contact, which makes sensitivity calculation very efficient. Numerical implementation of the proposed shape and configuration DSA method is performed using the meshfree method. The accuracy and efficiency of the proposed method are illustrated by minimizing the springback in a benchmark S‐rail problem. Copyright © 2007 John Wiley & Sons, Ltd.

Continuum‐based design sensitivity analysis and optimization of nonlinear shell structures using meshfree method

AbstractA continuum‐based shape and configuration design sensitivity analysis (DSA) method for a finite deformation elastoplastic shell structure has been developed. Shell elastoplasticity is treated using the projection method that performs the return mapping on the subspace defined by the zero‐normal stress condition. An incrementally objective integration scheme is used in the context of finite deformation shell analysis, wherein the stress objectivity is preserved for finite rotation increments. The material derivative concept is used to develop a continuum‐based shape and configuration DSA method. Significant computational efficiency is obtained by solving the design sensitivity equation without iteration at each converged load step using the same consistent tangent stiffness matrix. Numerical implementation of the proposed shape and configuration DSA is carried out using the meshfree method. The accuracy and efficiency of the proposed method is illustrated using numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

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.

Configuration design sensitivity analyss of nonlinear transient dynamics

A continuum-based configuration design sensitivity analysis (DSA) method is developed for the transient dynamic response of nonlinear structural systems. The first-order variations of energy forms, load form, and kinematic and structural responses with respect to the configuration design variables are derived. For the configuration design, both shape and orientation variations contribute to the first-order variation of the governing equation. A material derivative approach is employed to represent the shape variation. The shape and orientation design variations usually defined in element local coordinates are expressed in terms of global coordinates in this paper. A direct differentiation method is used since it is more appropriate for both path independent and path dependent problems than the adjoint variable method. For the structural domain, updated Lagrangian formulation is used so that the design velocity field that defines the mapping between initial and perturbed design velocity field that defines the mapping between initial and perturbed designs is updated at each updated configuration. The Hughes-Liu truss/beam element based on the degeneration of an 8-node isoparametric solid is selected to discretise the struuctural domain. The objective stress and strain measures and Jaumann rate-based incrementally objective stress integration scheme are employed to handle large rotation effects. In temporal domain, an explicitly central difference method is used for time integration. For accurate numerical implementation, a fixed time step, sufficiently smaller than the critical time step that is determined by element geometry and material properties, is selected to remove the effect of time step sensitivity on the sensitivity of the performance measures. The developed DSA method is implemented to validate the accuracy and efficiency of the proposed method, which yields very good agreement with the central finite difference results.

Configuration design sensitivity analysis for dynamic systems using CAD-based velocity field

Configuration design sensitivity analysis for dynamic systems using CAD-based velocity field

Configuration design sensitivity analysis of dynamics for constrained mechanical systems

A continuum-based configuration design sensitivity analysis method is developed for dynamics of multibody systems. The configuration design variables of multibody systems define the shape and orientation changes. The equations of motion are directly differentiated to obtain the governing equations for the design sensitivity. The governing equation of the design sensitivity is formulated as an overdetermined differential algebraic equation and treated as ordinary differential equations on manifolds. The material derivative of a domain functional is performed to obtain the sensitivity due to shape and orientation changes. The configuration design sensitivities of a fly-ball governor system and a spatial four bar mechanism are obtained using the proposed method and are validated against these obtained from the finite difference method.

Design sensitivity analysis and optimization of non-linear transient dynamics. Part II?configuration design

For configuration design of non-linear dynamic structures, such as crashworthiness design, a continuum-based configuration design sensitivity analysis (DSA) and optimization methods are developed. The same transient dynamic analysis method used in Part I of this paper is employed here. The elastic–plastic material and finite strain and rotation effect are considered. The first-order variations of the energy forms, load form and kinematic and structural responses with respect to configuration design variables are derived. For the configuration design, both the shape and orientation variations contribute to the first-order variation of the equations of motion. The angular design velocity field associated with Euler angles is used to represent the orientation variation to obtain accurate design sensitivity results. Like Part I of the paper, the updated Lagrangian formulation and direct differentiation method are used for DSA for the path-dependent problem. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs should also be updated at each configuration. Numerical implementation of configuration DSA and optimization is carried out for fixed time steps using DYNA3D and the modified feasible direction (MFD) method. It is observed that the proposed DSA method yields better sensitivity results than the finite difference method for the highly non-linear problem. Design optimization is carried out using the design sensitivity information.Copyright © 2000 John Wiley & Sons, Ltd.

Configuration Design Sensitivity Analysis of Kinematic Responses of Mechanical Systems*

A continuum-based configuration design sensitivity analysis method is presented for kinematically driven mechanical systems. Configuration design variable for mechanical systems are defined. The 3-1-3 Euler angle is used as the orientation design variable. The reassembly process for a mechanical system after a configuration design change is eliminated by imposing kinematic admissibility conditions of the velocity field. The direct differentiation method is used to derive design sensitivity equations. Numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.

Configuration Design Sensitivity Analysis of Nonlinear Structural Systems with Elastic Material*

ABSTRACT A continuum-based design sensitivity analysis (DSA) method is presented for configuration design of nonlinear structural systems using Mindlin plate and Tim-oshenko beam theories. Both displacement and critical load performance measures are considered. Configuration design variables are characterized by shape and orientation changes of structural components. The material derivative that is used to develop the continuum-based shape DSA method is extended to account for effects of configuration design variation. The piecewise linear design velocity field, i.e., C0-regular, is used to support configuration design changes for a broad class of built-up structures with beams and plates. To allow use of the C0-design velocity field, mathematical models of beam and plate bending must be second-order partial differential equations, so that only first-order derivatives appear in the integrand of the energy equation and, thus, in the integrand of the configuration design sensitivity expression. Since the Mindlin plate and Timoshenko beam theories use displacement and rotation to describe structural response, mathematical models of beam and plate bending are reduced to second-order partial differential equations. The isoparametric finite element formulations are used for numerical evaluation of continuum design sensitivity expressions.

Configuration design sensitivity analysis of built‐up structures part II: Numerical method

AbstractA numerical method is presented for structural configuration design sensitivity calculations using established finite element analysis codes. The theoretical foundation of the configuration design sensitivity analysis is given in Part I of this paper1 using the continuum elasticity formulation. A linear approximation between design parameterizations and design velocity fields is derived for line and surface design components. A regular design velocity that avoids the calculation of corner terms at the boundary is implemented to show a unified configuration design sensitivity analysis. The design sensitivity analysis method is demonstrated using the post‐processing data of an existing finite element code. Configuration design sensitivity results of displacement, stress and eigenvalue performance measures are illustrated for several built‐up engineering structures.

Configuration design sensitivity analysis of built‐up structures part I: Theory

AbstractA continuum based configuration design sensitivity analysis method is developed for built‐up structures that include truss, beam, plane elastic solid and plate design components. The configuration design variation of a structural component can be characterized by changes in the domain shape and in the orientation of the component. A variational approach is then used to incorporate both shape and orientation effects in the same energy equation. A continuum shape design sensitivity analysis method that utilizes the material derivative idea of continuum mechanics is employed to account for effects of shape variation. An approach that is similar to the continuum shape design sensitivity analysis method is developed in this paper to account for the effect of orientation variation. Variations of energy bilinear and load linear forms, with respect to both shape and orientation design variables, are derived for each structural component. Using the adjoint variable or direct differentiation method, configuration design sensitivity results for built‐up structures are obtained in terms of the design velocity fields. Configuration design sensitivity analyses of both static and eigenvalue responses are considered.

An Extension of the Material Derivative Method for Configuration Design Sensitivity Analysis

Abstract Configuration design variation of built-up structures that include truss, beam, plane elastic solid, and plate design components can be characterized by changes in the domain shape and orientation of design components. The material derivative method of shape design sensitivity analysis is extended to develop a continuum configuration design sensitivity analysis method for built-up structures by accounting for effects of both shape and orientation changes in the same energy equation. Variations of energy bilinear and load linear forms, with respect to both shape and orientation design variables, are derived for each structural component. Using the adjoint variable or direct differentiation method, configuration design sensitivity results for built-up structures are obtained in terms of the design velocity fields. The relationship between design parameterizations and design velocity fields is discussed for line and surface design components. A numerical method is developed to evaluate these configu...