Author(s): Kiran N. Solanki
Overview
Here, we present a methodology for simulationbased product
design optimization using an internal state variable (ISV) constitutive
modeling approach that captures the microstructureproperty relations in the
material. By modeling the stochastic uncertainties in the material model and
the loading conditions, the design optimization problem is formulated and
solved using the reliabilitybased design optimization (RBDO) methodology. The
application problem considers the design optimization of an A356T6 cast
aluminum component under maximum stress and damage constraints. Alternative
metamodeling techniques are used to develop appropriate surrogate models in
lieu of direct coupling of nonlinear static finite element analysis and
numerical design optimization. Probabilistic design constraints are modeled
using the safety index approach with the solution of the nested optimization
problem facilitated with the help of analytical surrogate models. Comparison of
the optimization results reveals the importance of using an ISVbased
constitutive model that is sensitive to the growth of damage in material.
Moreover, the solution of the RBDO problem captures the effects of uncertainty
on finding the minimum weight design for the cast component.
To illustrate the integration of microstructureproperty relations with RBDO
methodology, we consider the design of an automobile front suspension control
arm as depicted by its FE model in Figure. The control arm is made of A356
aluminum alloy, produced using the diecast process and then heat treated to T6
condition. This component has been selected because of the availability of data
on its loadfailure characteristics and the success of microstructureproperty
relationship model (Horstemeyer and Gokhale, 1999; Horstemeyer et al., 2000;
Horstemeyer, 2001)
^{ [1] }in the correct prediction of its failure mode. The two
critical loading conditions considered in the design are:
 a 0.8 g panic brake
 b 0.8 g pothole strike.
Both of which are modelled using appropriate boundary
conditions (generalized displacements) at the connection points A through F
shown in Figure. The FE model is comprised of 22,310 solid elements (C3D8) for
97,440 degrees of freedom. Nonlinear static simulations of this highfidelity
model are performed using the FE code, ABAQUSimplicit.
Two different design cases are considered. In the first
one, a deterministic optimization problem is set up and solved for minimum
weight using two different constitutive models whereas in the second one, an
RBDO problem is solved using the uncertainties associated with the loading
condition as well as those in the microstructure property relations in the
ISVdamage model.
Two different design cases are considered. In the first
one, a deterministic optimization problem is set up and solved for minimum
weight using two different constitutive models whereas in the second one, an
RBDO problem is solved using the uncertainties associated with the loading
condition as well as those in the microstructure property relations in the
ISVdamage model.
In summary, the steps in formulating and solving the deterministic optimization problem are as follows:
 Identify the DOE points (design variable values) for both the training and the test design points.
 Develop a morphed mesh corresponding to each design model in (1) using GENESIS.
 Perform a nonlinear FEA for each design model in (2) using ABAQUSimplicit under the specified loading conditions. Each analysis is performed using:
 simple plasticity model (standard ABAQUS material model)
 ISVdamage model (DMG 1.0 UMAT)
 Find the weight as well as the maximum φ and maximum von Mises stress for each design model.
 Identify the reasonable training points for each material model.

Develop six separate metamodels for each response corresponding to (3.1)
and (3.2). Evaluate the accuracy of the metamodels and identify the best
metamodel for each response.
 Solve the optimization problem using the selected metamodels.
Conclusions
A methodology for integration of computational material models offering accurate
microstructureproperty relations with simulationbased design optimization
under uncertainty was presented. The successful implementation of the
methodology required multiscale material modelling, uncertainty quantification
in microstructureproperty relations, nonlinear static FEA, mesh morphing,
development and evaluation of multiple surrogate models, structural reliability
analysis for the calculation of safety index and mathematical programming for
solution of deterministic and reliabilitybased optimization problems. The
presented methodology provides a means to quantify the tradeoff between
component geometry and material microstructure. In essence, the methodology
allows a designer to optimize the component for strength and/or weight while
concurrently developing criteria for tailored microstructures or new materials
that will further extend the component capacity. Moreover, the present
framework allows a designer to analyze stochastic microstructure distribution
effects on the uncertainty of the macroscale mechanical properties. Through the
solution of an application problem focusing on the design optimization of an
A356T6 cast aluminum component, two objectives were demonstrated:
 the advantage of using an ISVdamage model over the standard plasticity model for accurate prediction of damage.

the quantification of stochastic uncertainties at the
microstructure level and their propagation to macroscopic responses for use in
reliabilitybased design optimization.
Based on the solution of the application problem, the following conclusions can be drawn:
 When using the damage/failure criterion, the design variables had a highly nonlinear and noisy effect.
 Not all metamodels can accurately capture a noisy and nonlinear response, especially when the design and random variables combined are used as input variables.
 Mesh morphing was effective in altering the component geometry and sizing of a new structural component.

Due to its smaller damage level, the optimum design
based on ISVdamage constitutive model was found to be superior to that based
on simple plasticity models for the same weight.

Although a safety factor of two for load resulted in a safety factor of
approximately two for stresses, it led to a much greater safety factor in terms
of damage. Hence, the load safety factor must be selected with caution.

In comparison to the baseline model, the control arm design optimization resulted in
a weight saving of approximately 25% while increasing component safety by an order of
magnitude. One could effectively save more weight if the damage levels were designed
to be similar.
Resources

↑
Solanki, K.N., Acar, E., RaisRohani, M., Horstemeyer, M., & Steele, G. (Oct 2009). Product Design Optimization
with MicrostructureProperty Modeling and Associated Uncertainties. International Journal of Design Engineering Inderscience Publishers, 2(1), 4779.
(http://www.inderscience.com/search/index.php?action=record&rec_id=28446)