A novel approach for risk-averse structural topology optimization under uncertainties is presented, which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is assumed. For practical purposes it is important to design structures which are also robust to infrequent events. Hence, instead of an optimization with respect to the expectation of the involved random fields, we employ the Conditional Value at Risk (CVaR) in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is numerically challenging, a representation in the modern hierarchical tensor train format is proposed. In order to obtain such an efficient representation of the solution of the random state equation, a tensor completion algorithm is employed, which only requires the pointwise evaluation of solution realizations and can thus be considered nonintrusive. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach.
Computationally efficient handling of successive analyses required in structural design optimization under uncertainty
Prof. Dimos Charmpis | University of Cyprus | Cyprus
Reliability-Based Design Optimization (RBDO) and Robust Design Optimization (RDO) are the two most common approaches applied to structural design optimization problems under uncertainty. RBDO is a single-objective optimization procedure that minimizes the weight or cost of the structure under a pre-specified constraint on the failure probability of the final design. RDO, on the other hand, is a multi-objective optimization approach, which focuses at minimizing both the weight/cost of the structure and the sensitivity/variability introduced in the structural response of the final design due to input uncertainties.
The optimization algorithms (mathematical programming, genetic algorithms, etc.) employed nowadays to implement RBDO and RDO approaches, as well as the utilized procedures for reliability/variability estimations (e.g. various Monte Carlo simulation-based methods), involve expensive computations due to successive linear/nonlinear/eigenvalue/dynamic analyses required. The enormous computational burden associated with such multiple large-scale analyses may be substantially reduced by appropriately handling the most demanding tasks in terms of processing power and storage space needs: the successive systems of linear equations with multiple left- and/or right-hand sides that need to be solved. For this purpose, customized versions of iterative solution methods are presented, which are equipped with appropriate preconditioning techniques to accelerate convergence during successive solutions.
Large-scale RBDO and RDO examples involving multiple linear/nonlinear/eigenvalue/dynamic analyses are used to show that such demanding problems can be handled in a computationally efficient manner. It is demonstrated that, with the appropriate know-how, the use of RBDO and RDO in real world cases is computationally feasible despite the enormous computing requirements induced.
Risk average optimal control problem for elliptic PDEs with uncertain coefficients
We consider a risk averse optimal control problem for an elliptic PDE with uncertain coefficients. The control is a deterministic distributed forcing term and is determined by minimizing the expected L2-distance between the state (solution of the PDE) and a target deterministic function. An L2-regularization term is added to the cost functional.
We consider a finite element discretization of the underlying PDE and derive an error estimate on the optimal control.
Concerning the approximation of the expectation in the cost functional and the practical computation of the optimal control, we analyze and compare two strategies.
In the first one, the expectation is approximated by either a Monte Carlo estimator or a deterministic quadrature on Gauss points, assuming that the randomness is effectively parametrized by a small number of random variables. Then, a steepest descent algorithm is used to find the discrete optimal control.
The second strategy, named Monte Carlo Stochastic Approximation is again based on a steepest-descent type algorithm. However the expectation in the computation of the steepest descent is approximated with independent Monte Carlo estimators at each iteration using possibly a very small sample size. The sample size and possibly the mesh size in the finite element approximation could vary during the iterations. We present error estimates and complexity analysis for both strategies and compare them on few numerical test cases.
Two Step Uncertainty Quantification Using Gradient Enhanced Stochastic Collocation for Geometric Uncertainties
Uncertainty quantification(UQ) for Fluid-Structure Interaction problems is challenging in terms of the computational cost, time and efficiency. Sampling algorithms like Monte-Carlo(MC) are not feasible for these problems due to constraints in computational time and cost. Random geometry variation arises due to manufacturing tolerances, icing phenomenon, wear and tear during operation etc. Quantification of these geometric uncertainties is challenging due to large number of geometric parameters resulting in a high dimensional stochastic problem.
A two-step UQ using the gradient information obtained by solving the adjoint equation is employed in the current study. A gradient enhanced stochastic collocation(GESC) using polynomial chaos(PC) approach is used. The uncertainty in geometry is represented by Karhunen-Loeve expansion with predefined covariance function. In first step, using the sensitivity information, the parameters that don't have major influence on output quantity of interest(QoI) are identified. A reduction in dimension of random space is achieved by setting these parameters as deterministic. The QoI is represented by a PC representation. A set of collocation points is selected on the stochastic domain and the FEM model along with the adjoint equation is solved at each of these points to obtain QoI and gradient of QoI with respect to each of the uncertain input parameters. The stochastic collocation(SC) strategy is modified to incorporate the additional gradient information. The deterministic coefficients in the PC expansion of the QoI are determined by a least square regression of system of equation that contains QoI and gradients of QoI. The method is tested for a cylinder in flow with uncertain geometric perturbations and uncertainties in drag(QoI) is evaluated. The accuracy is compared with SC and MC. The method is computationally more efficient than SC and MC. The method can be extended for shape optimization problems considering the uncertainty in geometry.
A clustering method for uncertainty propagation with dependent inputs
Prof. Daan Crommelin | CWI Amsterdam | Netherlands
For uncertainty propagation with multivariate inputs, a basic assumption
underlying many methods is that the elements of the input vector are
mutually independent. We propose a new method, based on clustering, for
cases where the inputs are dependent. In this method, the cluster
centers and associated cluster sizes are used as nodes and weights for a
quadrature rule by which moments of the model output can be efficiently
estimated. The computational cost of determining the centers and weights
is small. The clustering approach can be used for non-Gaussian inputs as
well as for situations where the distribution of the inputs is unknown
and only a sample of inputs is available. No fitting of the input
distribution is needed.
We demonstrate the performance of the clustering method using test
functions and a CFD benchmark case (lid-driven cavity flow). Tests with
input dimension up to 16 are included, showing strong performance in
tests with high correlation between inputs.