Publications

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When solving partial differential equations (PDEs) with random inputs, it is often computationally inefficient to merely propagate samples of the input probability law (or an approximation thereof) because the input law may not accurately capture the behavior of critical system responses that depend on the PDE solution. To further complicate matters, in many applications it is critical to accurately approximate the “risk” associated with the statistical tails of the system responses, not just the statistical moments. In this paper, we develop an adaptive sampling and local reduced basis method for approximately solving PDEs with random inputs. Our method determines a set of parameter atoms and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. In addition, we extend our adaptive sampling approach to accurately compute measures of risk evaluated at quantities of interest that depend on the PDE solution.

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In this report we formulate eigenvalue-based methods for model calibration using a PDE-constrained optimization framework. We derive the abstract optimization operators from first principles and implement these methods using Sierra-SD and the Rapid Optimization Library (ROL). To demon- strate this approach, we use experimental measurements and an inverse solution to compute the joint and elastic foam properties of a low-fidelity unit (LFU) model.

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In this report we derive frequency-domain methods for inverse characterization of the constitutive parameters of viscoelastic materials. The inverse problem is cast in a PDE-constrained optimization framework with efficient computation of gradients and Hessian vector products through matrix free operations. The abstract optimization operators for first and second derivatives are derived from first principles. Various methods from the Rapid Optimization Library (ROL) are tested on the viscoelastic inversion problem. The methods described herein are applied to compute the viscoelastic bulk and shear moduli of a foam block model, which was recently used in experimental testing for viscoelastic property characterization.

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