Continuum Sensitivity Method Research Articles

Multi-scale computational modeling of lightweight aluminum-lithium alloys

The present study addresses the multi-scale computational modeling of a lightweight Aluminum-Lithium (Al-Li) 2070 alloy. The Al-Li alloys display significant anisotropy in material properties because of their strong crystallographic texture. To understand the relationships between processing, microstructural textures at different material points and tailored material properties, a multi-scale simulation is performed by controlling the texture evolution during deformation. To achieve the multi-scale framework, a crystal plasticity model based on a one-point probability descriptor, Orientation Distribution Function (ODF), is implemented to study the texture evolution. Next, a two-way coupled multi-scale model is developed, where the deformation gradient at the macro-scale integration points is passed to the micro-scale ODF model and the homogenized stress tensor at the micro-scale is passed back to the macro-scale model. A gradient-based optimization scheme which incorporates the multi-scale continuum sensitivity method is utilized to calibrate the slip system parameters of the alloy using the available experimental data. Next, the multi-scale simulations are performed for compression and tension using the calibrated crystal plasticity model, and the texture data is compared to the experiments. With the presented multi-scale modeling scheme, we achieve the location-specific texture predictions for a new generation Al-Li alloy for different deformation processes.

Control of macrosegregation during the solidification of alloys using magnetic fields

Numerical modeling of convection damping and macrosegregation suppression during solidification of alloys with prominent mushy zones through the use of tailored magnetic fields is demonstrated here. Macrosegregation leads to commonly observed defects such as freckles, channels and segregates in cast alloys that severely affect the performance and suitability of the alloy for further applications. The current work demonstrates the successful use of magnetic fields in suppressing thermosolutal convection and eliminating some of these defects in solidifying metallic alloys. The computational model presented utilizes volume-averaged governing transport equations and stabilized finite element techniques to discretize these equations. A finite-dimensional optimization problem, based on the continuum sensitivity method is considered to design the time history of the imposed magnetic field required to effectively damp convection. The coefficients that determine this time variation are the main design parameters of this optimization problem. Continuum sensitivity equations are derived by design-differentiating the governing equations of the direct problem. The cost functional here is given by the square of the L 2 norm of an expression representing the deviation of the volume-averaged velocity corresponding to conditions of convection less growth. The cost functional minimization process is realized through a non-linear conjugate gradient algorithm that utilizes finite element solutions of the continuum direct and sensitivity problems. Design of the time history of the imposed magnetic field is highlighted through different examples with the main objective being the suppression of convection and macrosegregation during alloy solidification.

Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization

Evolution of properties during processing of materials depends on the underlying material microstructure. A finite element homogenization approach is presented for calculating the evolution of macro-scale properties during processing of microstructures. A mathematically rigorous sensitivity analysis of homogenization is presented that is used to identify optimal forging rates in processes that would lead to a desired microstructure response. Macro-scale parameters such as forging rates are linked with microstructure deformation using boundary conditions drawn from the theory of multi-scale homogenization. Homogenized stresses at the macro-scale are obtained through volume-averaging laws. A constitutive framework for thermo-elastic–viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modelling microstructure evolution. The continuum sensitivity method (CSM) used for designing processes involves differentiation of the governing field equations of homogenization with respect to the processing parameters and development of the weak forms for the corresponding sensitivity equations that are solved using finite element analysis. The sensitivity of the deformation field within the microstructure is exactly defined and an averaging principle is developed to compute the sensitivity of homogenized stresses at the macro-scale due to perturbations in the process parameters. Computed sensitivities are used within a gradient-based optimization framework for controlling the response of the microstructure. Development of texture and stress–strain response in 2D and 3D FCC aluminum polycrystalline aggregates using the homogenization algorithm is compared with both Taylor-based simulations and published experimental results. Processing parameters that would lead to a desired equivalent stress–strain curve in a sample poly-crystalline microstructure are identified for single and two-stage loading using the design algorithm.

The continuum sensitivity method for the computational design of three-dimensional deformation processes

Although simulation of metal forming processes using modern computational tools has reached an advanced stage, there is still a great need to develop computational techniques for process design. In this work, the continuum sensitivity method (CSM) is extended to three-dimensions (3D) to accurately compute sensitivity fields and in particular changes in the design objectives as a result of infinitesimal perturbations to the design parameters. CSM involves differentiation of the governing field equations of the direct problem (constitutive, contact and kinematic problems) with respect to the design variables and development of the weak forms for the corresponding sensitivity equations. The present 3D developments include a novel regularized approach to the contact sensitivity problem that addresses the non-differentiability of the contact constraints. The computed sensitivity fields are used in a gradient-based optimization framework for process design (optimization) in 3D metal forming applications. A typical design problem at each optimization iteration involves simultaneous solution of a direct problem and a number of sensitivity problems corresponding to each of the design variables. Relevant 3D design problems related to preform and die design for desired properties in the final product are considered highlighting the features and potential of the metal forming design simulator developed.

Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties

Development of techniques for the control of material properties of polycrystal materials that are inherently dependent on preferred orientations (texture) is addressed. To account for the infinite degrees of freedom of microstructural features, a model reduction on the micro-scale is introduced. Reduced-order models are developed to model the evolution of microstructure described by an orientation distribution function using a finite element discretization of the orientation space. This reduced-order modeling approach is based on the technique of proper orthogonal decomposition (POD) and the method of snapshots. Furthermore, novel design problems are introduced for the control of microstructure based on realistic polycrystalline plasticity. Specifically, a gradient based optimization framework is introduced using a multi-length scale continuum sensitivity method (CSM). The model reduction is extended to the sensitivity analysis and is a key element for the success of computational design of deformation processes. Numerical examples that highlight the benefits of the continuum sensitivity method and model reduction are presented. In addition, the potential of the presented techniques towards process design for obtaining desired material properties is demonstrated with the control at a material point of the elastic modulus in fcc copper. A desired distribution of the elastic modulus is achieved through an optimal selection of the velocity gradient.

Deformation process design for control of microstructure in the presence of dynamic recrystallization and grain growth mechanisms

The idea of control of mechanical properties in materials through designer processing techniques is promoted through this effort, in contrast to the standard control by alloying. Microstructure evolution through dynamic recrystallization is accurately modelled for the large thermo-mechanical deformation of hyperelastic thermo-viscoplastic materials. An innovative solution strategy and a computational algorithm are presented for the solution of the direct deformation problem comprised of the kinematic, constitutive, contact and thermal sub-problems. The description relies on microstructure based scalar state variables. The formulation follows an approach introduced earlier by Busso [International Journal of Plasticity 14(4–5) (1998) 319] that considers an instability criterion to define the onset of recrystallization. The analysis is performed for hot forming processes and the kinetics of grain growth is controlled by the grain boundary energies and the energy stored through the dislocation density. A finite element implementation is presented and validated with available numerical and experimental results. In addition, an innovative computational framework is developed for the design of deformation processes using the continuum sensitivity method (CSM) incorporating microstructure related state variables. This CSM involves considering the direct continuum equations, design-differentiating them and then evaluating their discretized form. This method differs from common sensitivity techniques in which one design-differentiates the direct discretized equations. The developed method is validated by comparing the computed sensitivities to those obtained through a finite difference scheme. In addition, a gradient-based optimization framework is developed which uses the continuum sensitivity method to evaluate the gradients of the objective functions and constraints. The effectiveness of the method is demonstrated through a number of applications in the design optimization of hot forming processes in the presence of recrystallization and grain growth mechanisms. These examples cover a wide range of applications as they include parameter design (e.g., die design), preform design as well as multi-stage forming design.

A continuum sensitivity method for the design of multi-stage metal forming processes

A novel, efficient and mathematically rigorous continuum based sensitivity method is introduced that can be used to accurately evaluate the gradients of the objective function and constraints in the optimization-based design of multi-stage deformation processes. Weak sensitivity equilibrium equations are derived for the large deformation of the workpiece in each forming operation. This sensitivity kinematic problem is linearly coupled with the corresponding continuum sensitivity constitutive, contact and thermal sub-problems for the particular process. Thus a linear sensitivity problem with appropriate driving forces is identified and the analysis is carried out in an infinite dimensional framework. The multi-stage continuum sensitivity analysis takes a form similar to the updated Lagrangian sensitivity framework developed earlier for the design of single-stage deformation processes. It allows us to treat in a unified manner shape and parameter sensitivity analyses that are both present in a typical design problem of multi-stage deformation processes. The effectiveness of the proposed methodology is demonstrated with the solution of three practical problems in the design of two-stage metal forming processes.

Computational design of deformation processes for materials with ductile damage

A continuum sensitivity method is developed for thermoplasticity combined with ductile damage at finite strains. The computed sensitivity fields are used within a gradient-based optimization framework for the computational design of metal forming processes for porous materials. The accuracy and effectiveness of the developed updated Lagrangian finite element analysis and design techniques are demonstrated with a number of representative examples. In addition to die design problems, novel preform (shape) design problems are examined for near net shape manufacturing that accounts for the volume change induced during the deformation process.

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

AbstractA computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.

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...