Accuracy enhancement of fatigue damage counting using design sensitivity analysis

Three methods for design sensitivity such as numerical differentiation, analytical method and semianalytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semianalytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

Several new formulations for structural design sensitivity analysis (DSA) of elastic solids are presented that consistently account for steady and transient thermal loading, along with centrifugal and gravity loading. The approach employs implicit differentiation of the discretized boundary integral equations and facilitates the production of accurate and efficient computational procedures that obviate the need for matrix factorizations in the DSA step. For centrifugal, gravity and general thermal loading, a new DSA concept employing a particular integral technique is developed. For centrifugal, gravity and steady thermal effects, a surface integration approach is also presented, and the two methods compared. Alternative strategies for modelling the behaviour of temperature distributions of evolving shapes undergoing shape optimization and the ramifications of these strategies in the DSA process are also discussed. It is shown that, for certain classes of thermoelastic problems, shape optimization can be performed without the need to incorporate thermal analysis or DSA into the overall process. A thermoelastic surface stress sensitivity analysis formulation is presented that does not require additional numerical integration. A new thermoelastic sharp corner traction component consistent sensitivity analysis formulation is also presented. This technique is shown to work for any initial stress distribution. Numerical sensitivity results are presented for many problems with analytical solutions.

Design sensitivity analysis and topology optimization of displacement–loaded non-linear structures

Improved semi-analytic sensitivity analysis combined with a higher order approximation scheme in the framework of adjoint variable method

The one- and two-dimensional inviscid Euler equations are formulated in an integro-differential form for shape design sensitivity analysis and optimization. The principal tool employed to derive the performance derivative sensitivity equations is the variational method, which is a continuous alternative to the discrete sensitivity analysis. Along with the sensitivity equations, the co-state equations and their boundary conditions are derived. Thereafter, based on the modal analysis of Von Neumann theory, the stability limits of the co-state equations are investigated. The stability characteristics of the co-state equations are compared with those of the Euler equations. Finally, using the criteria from the stability analysis of the state and co-state equations, some results of shape design optimization and sensitivity analysis for both quasi one- and two-dimensional Euler equations are presented. Copyright © 1999 John Wiley & Sons, Ltd.

Design Sensitivity Analysis (DSA) is a procedure for obtaining derivatives of design objectives, such as mass and strength with respect to some design parameters, such as size and shape. In the DSA methodology, we take an existing technique for computing objectives from parameters and systematically differentiate it so that the derivatives of the objectives can be computed directly, rather than employing some sort of differencing scheme to approximate derivatives. In this paper, we apply this technique to a non-linear finite element program, Radioss, and begin looking at the properties of the DSA methodology for crashworthiness simulations. It is shown that some major computational advantages can be gained by employing DSA for these kinds of problems. Besides cost savings, there are many cases where numerical differentiation results in erroneous derivatives due to numerical noise in crash simulations. It is shown that the DSA method avoids these difficulties, thus, making it quite important for use in support of design activities.

In this paper, continuum-based sizing and configuration design sensitivity analysis (DSA) and optimization methods using the established transient dynamic analysis code DYNA3D are developed. For transient dynamic analysis, the explicit central difference method is used. For the structural domain, the updated Lagrangian formulation is used since it is appropriate for the path dependent problem. The first-order variations of the kinetic energy form, strain energy form, load form, and kinematic and structural responses with respect to the sizing and configuration design variables are derived. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs is updated at each updated configuration. The direct differentiation method is used in this paper for DSA. Numerical implementation of sizing DSA, configuration DSA, and optimization of a vehicle frame and A-pillar models has been carried out using the nonlinear finite element analysis (FEA) code DYNA3D.

An overview of the main approaches to sizing and shape design sensitivity analysis of linear and non-linear structural systems is presented. Recent conclusions regarding computational cost, accuracy, and ease of implementation are described. Implementation of design sensitivity analysis using existing finite element computer codes is discussed. It is shown how design sensitivity computations can be distributed over a network of computers to speedup computations in support of an interactive design environment.

A continuum-based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non-linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non-linear dynamic problems, such as crashworthiness design. The first-order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path-dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate-based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.

Using the direct differentiation method, a design sensitivity analysis method for time-dependent incompressible fluids is developed. The fluid behavior is described as the motion of particles involved by the SPH method. In the SPH projection method, instead of changing the fluid density, incompressibility is enforced by the pressure Poisson equation derived from pressure projection, which enable to use larger time steps. In spite of the additional pressure Poisson equation, the computational cost for the design sensitivity is not expensive since the factorized system matrix of pressure Poisson equation can be utilized. Aforementioned computational efficiency is very beneficial for the design sensitivity computation required for every time step in explicit time integration and updated Lagrangian schemes, for which an update scheme of design velocity field is developed using the velocity sensitivity. Through demonstrative numerical examples, the developed DSA method turns out to be efficient and shows excellent agreement with finite differencing.

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

Design sensitivity analysis for parameters affecting geometry, elastic–viscoplastic material constant and boundary condition by consistent tangent operator-based boundary element method

A method for the prediction of strain responses in a linear beam system using only output data—without direct measurement of stress or strain—is suggested. An indirect method is evaluated for predicting the strain behavior at a “hot spot” on a notched simple beam using the frequency response function (FRF) relating strain and acceleration. The dynamic response of the simple beam is obtained through uniaxial excitation. The results show that the prediction of dynamic strain is accurate within an error of less than 10% as compared to the measured strain data, indicating that the FRF prediction method is reliable enough to be used in practical applications.

Methodologies for greatly improving machine structural dynamics are presented in which design sensitivity analyses and evaluative parameters are used according to the fundamental procedures of the design optimization method based on clarification of competitive and cooperative relationships between characteristics. First, design sensitivity coefficients and evaluative parameters of structural dynamics are described. Then, design improvement methods of structural dynamics are proposed for the following three cases : (1) addition of elastic structural members, (2) addition of mass elements, and (3) substantial changes of joint design variables. Cases (1) and (2) correspond to the changes of the initial framework or configuration, and case (3) corresponds to the alteration of poor initial design variables. Finally, the methods are demonstrated with numerical examples for a machine tool structural model.