Design Sensitivity Analysis Method Research Articles

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.

Shape Design Optimization of Thermoelastic Structures for Durability

Shape design sensitivity analysis (DSA) and optimization methods for the fatigue life of thermoelastic structural components are presented in this paper. A multiaxial fatigue life prediction method is used for crack initiation. The crack initiation life prediction is modeled using constant amplitude strain-life data and cyclic stress-strain curves. A hybrid DSA method is used for the fatigue life. The design sensitivities of the dynamic stress and the temperature field are obtained using analytical approaches. The design sensitivity is used to predict the dynamic stress and temperature of the perturbed design. Using predicted stress and temperature, the fatigue life of the perturbed structural design component is predicted. The predicted fatigue life is then used to obtain the design sensitivity of the fatigue life by utilizing the finite difference method. The proposed DSA method is applied to design optimization of an automotive exhaust manifold of an automotive vehicle, considering crack initiation lives as design constraints.

Design sensitivity analysis method for multidisciplinary shape optimisation problems with linear and non‐linear responses

A method to solve shape and size optimisation problems with linear and non‐linear responses has been studied taking advantage of statistical methodologies. A nested optimisation procedure has been fixed. The global optimisation problem is decomposed in several subproblems where each non‐linear response is locally approximated with a first degree polynomial function identified by the definition and execution of an experimental plan. The approximating functions so obtained are used to evaluate the design sensitivity coefficients required by the optimisation procedure. The numerical results obtained during the optimisation process to verify exactly the value of the non‐linear responses are used to verify and to improve the approximating function accuracy. The non‐linear design sensitivity analysis method so defined has been used to solve a multidisciplinary shape optimisation problem involving a real 3D automotive structure.

Design Sensitivity Analysis of Hyperelastic Structures Using a Meshless Method

A continuum-based design sensitivity analysis method for hyperelastic structures is presented. Analysis is performed using a meshless method, called the reproducing kernel particle method. Yeoh’ s energy density function is used to describe thehyperelastic structural behavior. The meshless method eliminates mesh distortion orentanglement encountered in using e nite element analysis for large deformation structural analysis and structural shape design optimization. Both the adjoint variable and direct differentiation methods are developed for material and shape design variables. An ine nitely long rubber tube, a two-dimensional rubber band, and an engine mount are examples used to demonstrate the feasibility and accuracy of the method.

Explicit and Implicit Methods for Design Sensitivity Analysis of Nonlinear Structures Under Dynamic Loads

Explicit and implicit methods for design sensitivity analysis of nonlinear structures subjected to transient dynamic loads are described. The direct differentiation method is used to calculate the sensitivities. Elastic and elasto-plastic problems undergoing large deformation are treated in the formulation. The explicit method is quite straightforward to implement for analysis as well as design sensitivity analysis: knowing all the quantities at time t, one updates the quantities for the next time point without solution of any linear system of equations, or any iterations. For the implicit method, the effective stiffness matrix for the sensitivity equation is different from that for the analysis phase due to inclusion of large deformation in the formulation. A numerical procedure to integrate the equation is proposed which uses the symmetric effective stiffness matrix from the analysis phase. Although the sensitivity analysis problem is linear, the procedure requires iterations for sensitivity calculations at each time step, as for the analysis phase. The implicit method is implemented using the pre-conditioned conjugate gradient iterative solution scheme. Numerical results for sensitivities indicate the implicit method to be more accurate than the explicit method for transient elastoplastic problems. However, the explicit method is substantially more efficient than the implicit method.

An efficient approach to structural optimization

It is now widely recognized that structural optimization using mathematical programming techniques can be employed efficiently only in conjunction with explicit approximate models. In this work an efficient optimization methodology combining a finite element-based approximate analysis model, a sequential quadratic programming algorithm incorporating an active set strategy and a direct method of design sensitivity analysis is developed. The methodology involves the solution of a sequence of explicit high-quality approximate problems subject to given move limits in the design space. A new technique for constructing approximation functions with a high quality adaptive capability to the original functions is proposed by using the values of the state variables (displacements and/or stresses) and their derivatives at points obtained in the process of optimization. Other approximation techniques have been presented and comparisons have been made using real-life automobile structures to demonstrate the power and generality of the approximation concepts in structural optimization.

2-D and 3-D shape optimization using mesh velocities to integrate analytical sensitivities with associative CAD

The design sensitivities generated with the mesh velocity method, used by the authors, are compared with those obtained by the boundary layer and boundary displacement methods. The effect of adaptive mesh refinement and error control on the quality of the velocity fields is discussed, as well as their ability to yield accurate first-order predictions of constraint values. Two numerical shape optimization examples of a 2D and a 3D component are presented. These examples are used to illustrate the benefits of integrating analytical methods of design sensitivity analysis with parametric capabilities supported by state-of-the-art CAD systems.

SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION FOR STRUCTURAL DURABILITY

SUMMARY In this paper, a design sensitivity analysis (DSA) method for fatigue life of 3-D solid structural components of mecanical systems with respect to shape design parameters is presented. The DSA method uses dynamic stress DSA obtained using an analytical approach to predict dynamic stress increment due to design changes; computes fatigue life of the component, including crack initiation and crack propagation, using the predicted dynamic stress; and uses the di⁄erence of the new life and the original life at the same critical point to approximate the sensitivity of fatigue life. A tracked vehicle roadarm is presented in this paper to demonstrate accuracy and eƒciency of the DSA method. Also, this method is applied to support design optimization of the tracked vehicle roadarm considering crack initiation lives as design constraints. ( 1997 by John Wiley & Sons, Ltd.

Design Sensitivity Analysis of Structure-Induced Noise and Vibration

A continuum design sensitivity analysis (DSA) method for dynamic frequency responses of structural-acoustic systems is developed using the adjoint variable and direct differentiation methods. A variational approach with a non-self-adjoint operator for complex variables is used to retain the continuum elasticity formulation throughout derivation of design sensitivity results. It is shown that the adjoint variable method is applicable to the variational equation with the non-self-adjoint operator. Sizing design variables such as the thickness and cross-sectional area of structural components are considered for the design sensitivity analysis. A numerical implementation method of continuum DSA results is developed by postprocessing analysis results from established finite element analysis (FEA) codes to obtain the design sensitivity of noise and vibration performance measures of the structural-acoustic systems. The numerical DSA method presented in this paper is limited to FEA and boundary element analysis (BEA) is not considered. A numerical method is developed to compute design sensitivity of direct and modal frequency FEA results. For the modal frequency FEA method, the numerical DSA method provides design sensitivity very efficiently without requiring design sensitivities of eigenvectors. The numerical method has been tested using passenger vehicle problems. Accurate design sensitivity results are obtained for analysis results obtained from established FEA codes.

Shape Design Sensitivity and Optimization Using P-Version Design Modeling and Finite Element Analysis*

ABSTRACT A methodology that supports shape design modeling, structural shape design sensitivity analysis (DSA), and optimization of elastic solids, using p-version finite element analysis (FEA), is developed. The p-version FEA approach is attractive for shape design optimization, due to its high accuracy of analysis results, even with coarse mesh; ease of creation of design and finite element models; consistency between design and finite element models; insensitivity to finite element mesh distortion and aspect ratio; and tolerance for large shape changes during design optimization iterations. A continuum-based DSA method that is applicable to both planar and spatial structures is developed with an established p-version FEA code, STRESS CHECK. A shape design parameterization method that uses the geometric entities of STRESS CHECK is proposed and implemented. In STRESS CHECK, the shape design model is identical to the finite element model, so finite element modeling errors are eliminated. Design optimization is carried out by integrating the shape design parameterization, design velocity field computation; first-order shape design sensitivity computation, STRESS CHECK; and an optimization code, DOT. Examples are run successfully in full-batch mode to demonstrate the feasibility of the proposed integrated optimization method

Three-Dimensional shape optimization approach based on natural design variables and the boundary element method

This paper deals with shape optimization of three-dimensional elastic bodies. The proposed approach is based on natural design variables and shape functions and utilizes the boundary element method. The design variables are the magnitudes of a set of fictitious loads applied on the structure. Shape optimal design problems based on minimum mean compliance, peak stress minimization and stress constraints are considered. A general method for shape design sensitivity analysis using the material derivative concept and adjoint variable method is used. The sensitivity formula for a general stress constraint imposed over the optimized boundary is derived. Optimal shapes for three-dimensional problems are presented to show numerical applications.

Design optimization of hip prosthesis of thick laminated composites by developing finite element method and sensitivity analysis

An investigation has been performed to develop a finite element method for the analysis of the behavior of complicated three-dimensional thick laminated composite structure, and the method has been applied to the design of hip prosthesis with nonuniform cross-sections. The developed method can accomodate the varying material properties layered within the element and allow the ply-drop-off along the element edges. The numerical results are compared with the analytical solution, and the results show that the number of elements can be reduced up to 80% within 5% error bounds by using the ply-drop-off element. A method of design sensitivity analysis has also been developed to find the optimal ply angles minimizing the deflection of thick laminated composite structures. The developed finited element method together with the design sensitivity analysis has been applied to design of hip stem, and a procedure for calculating the optimum ply angle is proposed to minimize the rotational micromotion which is one of causes of patients’ pain.

Sensitivity analysis and optimization of truss/beam components of arbitrary cross-section—I. Axial stresses

In this paper, a general method for detailed design and optimization of the design parameters of an arbitrary cross-section of a truss/beam design component is proposed. The method draws upon already existing computer aided design (CAD) parametric capabilities to parametrize a two-dimensional closed contour. Any complex composite cross-section made of two-dimensional closed contours can be assigned to a design component. The method proceeds by computing the centroid, the cross-sectional area, the moments of inertia referred to the arbitrary origin and to the centroid. The principal angle of inertia is used to redefine the orientation of the coordinate reference system for finite element performance analysis. During the design optimization process the cross-section shape changes according to the design parametrization defined during the formulation of the design problem. The change in shape causes change in the orientation of the principal axes, causing in turn critical stresses to occur at new vertex locations. The continuum method of design sensitivity analysis is used to compute design sensitivity information with respect to the cross-sectional design parameters. The method is shown to be well suited for integration with an existing CAD and FEM code.

Design Sensitivity Analysis of Truss Structures with Elastoplastic Material*

ABSTRACT A continuum-based design sensitivity analysis (DSA) method is presented for configuration (or layout) design of nonlinear structural systems with rate-independent elastoplastic material. Configuration design variables are characterized by shape and orientation changes of the structural component. A continuum-based shape DSA method that utilizes the material derivative of continuum mechanics is extended to account for effects of shape and orientation variations. The incremental analysis method, with updated Lagrangian formulation, is used to derive the design sensitivity for the nonlinear structural system. To derive the design sensitivity, incremental energy and load forms are utilized. The first variations of energy and load forms and the static response with respect to configuration design variables are described using the material derivative. Direct differentiation is utilized to obtain the first variation of the performance measure explicitly in terms of variations of configuration design var...

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.

DESIGN SENSITIVITY ANALYSIS OF NON-LINEAR RESPONSE FOR LARGE DEFLECTION FORCED VIBRATIONS OF BEAMS

A computational method for design sensitivity analysis of the non-linear forced vibrations response of beams is presented. The large deflection leads to geometric non-linearity. Longitudinal displacement and inertia are considered in the non-linear vibration formulation. As an application of this method, two numerical examples are given. The results show that the method proves to be effective.

A continuum approach for second‐order shape design sensitivity of elastic solids with loaded boundaries

AbstractA second‐order shape design sensitivity analysis (DSA) method applicable to the shape change on the loaded boundaries is derived for three‐dimensional linear elastic solids using a continuum method with the material derivative. The continuum method is also used to derive mixed second‐order variations of stress and displacement performance measures with respect to shape design variables and distribution of non‐conservative traction loads, and also with respect to shape design variables and material properties. A shape design acceleration field is defined for the second‐order shape design sensitivity. Both direct differentiation and hybrid methods are presented in this paper. A numerical method, which can be implemented using established finite element analysis (FEA) codes, is developed. The feasibility and accuracy of the proposed second‐order shape DSA method has been demonstrated by solving a structural example‐doubly curved arch dam.

Continuum Sensitivity Method for Reliability-Based Structural Design and Optimization*

ABSTRACT The computational cost of reliability analysis and optimization of medium/large structural systems is prohibitively high. One of the reasons for this situation is the use of inefficient methods, such as finite differences, for design sensitivity analysis. A significant increase in the efficiency of reliability computations is expected from the integration of reliability methods with the discrete and continuum methods of design sensitivity analysis. So far, only the discrete method has been used in the so-called stochastic finite element method. Integration of reliability computations with the continuum method of design sensitivity analysis is proposed in this paper as a new, powerful approach to the probabilistic design of large structural systems. Basic formulations and solution methods of reliability analysis, design sensitivity analysis of reliability indices, reliability-based optimization, and multiobjective reliability-based optimization are reviewed with focus on the sensitivity informatio...

Optimal positioning of heating lines in a compression molding die using the boundary element method

A shape optimization problem is formulated to determine the optimal position of heating lines in a compression molding die. The objective of the problem is that the die cavity surface would be maintained by a prescribed uniform temperature. A new boundary integral equation for the sensitivity of the temperature in terms of the hole position is derived using the method of shape design sensitivity analysis. The boundary element method is employed to analyze the temperature and its sensitivity field. The presented sensitivity formulation is incorporated in an optimization routine. To demonstrate a numerical implementation, an example problem arising in thermal design of a compression molding die is dealt with, showing the effectiveness and usefulness of the presented method.

Design sensitivity analysis of nonlinear mode spectrum for large amplitude forced vibrations of beams

A computational method for design sensitivity analysis of a nonlinear mode spectrum for forced vibrations of beams is presented. The large deflection leads to geometric nonlinearity. Longitudinal displacement and inertia are considered in the nonlinear vibration formulation. A numerical example is given to illustrate the validity of the proposed method.