Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

Two forms of rate-sensitive constitutive equations, additive and multiplicative, are examined in the analysis of sheet metal forming using the finite element method. Results are obtained for hemispherical punch stretching of an AK steel sheet material with various punch speeds. The computed results in thickness strain distributions and load-displacement curves are almost identical for the two constitutive laws at a low punch speed. However, the additive law provides better agreement in the thickness strain distributions with the experimental trends for high-speed forming.

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.

In this paper, a numerical procedure has been proposed to predict limit strains of textured sheets under equi‐biaxial tension by means of the MK model and taking into account of crystal rotation. Limit strains have been predicted for three common crystallographic orientations and an annealed aluminium sheet. The effect of crystal rotation on limit strains was shown to be significant. However, no general trend of the effect has been observed. It is argued that the effect is quite complicated and dependent on various factors such as the initial orientation or texture, groove orientation and the initial inhomogeneous factor. The incorporation of texture change in the analysis of sheet metal forming was found essential.

A simplified one-step inverse analysis of sheet metal forming is a suitable tool to simulate the bending forming since the deformation path of bending forming is an approximately proportion one. A fast spring-back simulation method based on one-step analysis is proposed. First, the one-step inverse analysis is applied to obtain the stress distribution at the final stage of bending. Then, the unloading to get a spring back is simulated by LS-DYNA implicit solver. These processes are applied to the unconstrained cylindrical bending and the truck member rail. The spring-back and member rail widths at the several key sections are compared with experimental ones. It is well demonstrated that the proposed method is an effective way to predict the spring-back by unloading after bending process.

The objective of this study is to examine two analytical methods for the design sensitivity analysis of optimal control problems and to develop the analyses necessary for the application of the methods to thermal systems. The two methods are the direct differentiation method and the adjoint variable method. Results for these methods are compared with those obtained using central finite difference approximations for design sensitivity analysis. The adjoint variable method has the potential to reduce significantly computational times for design sensitivity analysis while sacrificing only a small amount of accuracy. As the number of functional requiring design sensitivity analysis increases, however, the adjoint variable method becomes less attractive and the direct differentiation method becomes more attractive.

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.

A new approach toward reliability based design optimization (RBDO) is proposed based on the response surface augmented moment method (RSMM). In RSMM, the reliability analysis procedure based on design of experiments (DOE) is combined with the response surface method (RSM) for numerical efficiency. It utilizes the Pearson system with four statistical moments to calculate the failure probability, and the progressive update of the response surface facilitates the calculation of these four statistical moments. In this study, a semi-analytic design sensitivity analysis is performed in connection with RSMM for an efficient implementation of RSMM in RBDO. The sensitivity of failure probability with respect to the design variables is calculated by direct differentiation and finite difference method with the Pearson system. It is integrated into a mathematical programming for RBDO and applied to several test examples. It was demonstrated that the proposed method of RBDO based on RSMM is efficient and robust.

Analysis of Sheet Metal Forming of Interstitial-Free Steel and Extra-DeepDrawing Steel Using ABACUS

Three methods for design sensitivity analysis such as finite difference method(FDM), direct differentiation method(DDM) and adjoint variable method(AVM) are well known. FDM and DDM for design sensitivity analysis cost too much when the number of design variables is too large. An AVM is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation. Because the adjoint equation is independent of the number of design variables, an AVM is efficient for when number of design variables is too large. In this study, AVM has been extended to the eigenproblem of damped systems whose eigenvlaues and eigenvectors are complex numbers. Moreover, this method is implemented into a commercial finite element analysis program by means of the semi-analytical method to show applicability of the developed method into practical structural problems. The proposed method is compared with FDM and verified its accuracy for analytical and practical cases.

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.

Design sensitivity analysis of dynamic response of nonviscously damped systems

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.

The boom structure is a key component of giant boom cranes, and the stability-ensured topology optimization is critical to its lightweight design. The finite difference method, direct differentiation or adjoint method needs many time-consuming nonlinear analyses for this problem with a large number of design variables and constraints, and the last two methods are difficult to implement in off-the-shelf softwares. To overcome these challenges, this work first defines a global stability index to measure the global stability of the whole structure, and a compression member stability index to identify the buckling of compression members. Numerical and experimental verifications of these two stability indices are conducted by analyzing a simple three-dimensional frame. Next, the anti-buckling mechanism of boom structures is analyzed to develop the precedence order of freezing relative web members. The stability indices and the freezing measure are then utilized as a part of a novel Stability-Ensured Soft Kill Option (SSKO) algorithm, built upon the existing Soft Kill Option (SKO) method. The objective is to minimize the discrepancy between structural volume and predetermined target volume, while the global stability and stress are regarded as constraints. Lastly, the SSKO algorithm with different scenarios is applied to topology optimization problems of four-section frames and a ring crane boom; in both cases the consistent and stable topologies exhibit applicability of the proposed algorithm.

Forced expression of Hnf4a induces hepatic gene activation through directed differentiation

The objective of this work is to develop and implement efficient numerical procedures for gradient based shape optimization of steady, strongly coupled fluid–structure interaction problems involving turbulent flow. The governing equations are the Reynoldsaveraged Navier-Stokes equations combined with a large displacement formulation for the structure. The eddy viscosity is computed using either algebraic turbulence models or the two-equation k − ω model. The solution for state is obtained using finite element residual formulations based on a consistent weak formulation of the fluid–structure interaction problem. Due to the possible large displacements of the structure, the fluid mesh is updated using a modified elastic analogy. The resulting nonlinear equations are solved using an approximate Newton method, making it possible to do design sensitivity analysis (DSA) by the direct differentiation method in a very efficient way where the resulting sensitivity equations are solved by an incremental iterative method. Gradient based shape optimization is illustrated for finding the shape of both an infinitely stiff and a flexible valve. Such examples of semi-analytic DSA and gradient based shape optimization of strongly coupled fluid-structure interaction problems involving turbulent flow modeled using two-equation turbulence models have, at least to the knowledge of the authors, not been presented before in the literature and thus represent novel results.