DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION OF NONLINEAR TRANSIENT DYNAMICS1*

The present thesis describes the development of a computational method for the numerical simulation of compressible flows in aerodynamics. The main goal is to study high-order discretization schemes that can potentially deliver a better computational efficiency. The spatial part of the conservation equations is discretized with the finite difference method. In particular, Summation By Parts (SBP) finite difference operators and treatment of boundary conditions via Simultaneous Approximation Terms (SAT) have been investigated. An SBP finite difference operator is essentially a (high-order) centered finite difference scheme with a specific closure at the boundary. The SAT are penalty-like terms that enforce the boundary conditions weakly and are used to augment the SBP schemes. The combination of SBP operators with SAT boundary terms constitutes schemes that, for the linearized equations governing a smooth flow, are provably stable (thus convergent) schemes. The discretization in time is carried out with high-order schemes as well. Both implicit and explicit time integration schemes have been investigated. The system of non-linear equations that arises when using the implicit time integration schemes is solved via (a damped) Newton's method. On the target grid, the initial guess required for Newton's method is obtained by performing grid sequencing combined a series of explicit relaxation iterations on each grid level. The flow equations and the equation of the turbulence model (if present) are solved in a fully-coupled manner. Numerical simulations for inviscid flow, laminar flow, as well as simulations based on DNS, LES and RANS have been performed. Results show that the described computational method is particularly well-suited for smooth external flows about relatively simple geometries, solved with the DNS and LES approach. For these types of flows the gain in efficiency obtained by using high-order schemes is clear (more than a factor of 5). For RANS, however, the comparison with a state-of-the-art second order finite volume code suggests that solving the turbulence and the flow equations fully coupled with implicit schemes may not be the most efficient iteration strategy. On the other hand, the numerical simulations of non-smooth/internal flows highlighted the need of further development of the SBP-SAT framework.

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.

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.

This work presents the development of an explicit/implicit particle finite element method (PFEM) for the 2D modeling of metal cutting processes. The purpose is to study the efficiency of implicit and explicit time integration schemes in terms of precision, accuracy and computing time. The formulation for implicit and explicit time marching schemes is developed, and a detailed study on the explicit solution steps is presented. The PFEM remeshing procedures for insertion and removal of particles have been improved to model the multiple scales of time and/or space of the solution. The detection and treatment of the rigid tool contact are presented for both, implicit and explicit schemes. The performance of explicit/implicit integration is studied with a set of different two-dimensional orthogonal cutting tests of AISI 4340 steel at cutting speeds ranging from 1 m/s up to 30 m/s. It was shown that if the correct selection of the time integration scheme is made, the computing time can decrease up to 40 times. It allows us to affirm that the computing time of the PFEM simulations can be excessive due to the used time marching scheme independently of the meshing process. As a practical result, a set of recommendations to select the time integration schemes for a given cutting speed are given. This is intended to minimize one of the negative constraints pointed out by the industry when using metal cutting simulators.

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.

A coupled implicit-explicit time integration method for compressible unsteady flows

Stability of semidiscrete formulations for parabolic problems at small time steps

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 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.

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.

This study investigates the full penetration simulation of piles from the ground surface, focusing on frictional contact modeling without mesh distortion. To overcome issues related to mesh distortion and improve solution convergence, the Arbitrary Lagrangian–Eulerian (ALE) adaptive mesh technique was implemented within both explicit and implicit time integration schemes. The numerical model was validated against field experiments conducted at Bothkennar, Scotland, using the Imperial College instrumented displacement pile (ICP) in soft clay, where the soil behavior was effectively represented using the modified Cam-Clay model and the Mohr–Coulomb model. The primary objectives of this study are to evaluate the ALE method performance in handling mesh distortion; analyze the effects of soil–pile interface friction, pile dimensions, and various dilation angles on pile resistance; and compare the effectiveness of explicit and implicit time integration schemes in terms of stability, computational efficiency, and solution accuracy. The ALE method effectively modeled pile penetration in Bothkennar clay, validating the numerical model against field experiments. Comparative analysis revealed the explicit time integration method as more robust and computationally efficient, particularly for complex soil–pile interactions with higher friction coefficients.

An adaptive h‐version finite element procedure is presented for a 2D elastodynamic transient analysis using direct integration. The procedure modifies the spatial discretization, i.e. the mesh, in time so that the spatial discretization error can be controlled in each time instant. We analyse an a priori error estimate for the total energy which shows that an a posteriori error estimate developed by Zienkiewicz and Zhu for linear elastostatics can reasonably be extended to dynamic transient analysis. For the time integration, both explicit and implicit time integration schemes can be coupled with the procedure. However, the study in this paper is confined to the use of the Newmark scheme. Numerical examples are used to study the performance of the error estimate and to illustrate the adaptive procedure. Obtained numerical results indicate that: (i) the extension of the Zienkiewicz–Zhu estimate to dynamic transient analysis is useful; (ii) the presented adaptive procedure can, in an efficient way, provide a finite element solution with a user‐controlled accuracy.

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.

Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme

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