4,780 research outputs found
Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems
Stochastic physical problems governed by nonlinear conservation laws are
challenging due to solution discontinuities in stochastic and physical space.
In this paper, we present a level set method to track discontinuities in
stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed
function that vanishes at discontinuities, the iso-zero of the level set
problem coincide with the discontinuities of the conservation law. The level
set problem is solved on a sequence of successively finer grids in stochastic
space. The method is adaptive in the sense that costly evaluations of the
conservation law of interest are only performed in the vicinity of the
discontinuities during the refinement stage. In regions of stochastic space
where the solution is smooth, a surrogate method replaces expensive evaluations
of the conservation law. The proposed method is tested in conjunction with
different sets of localized orthogonal basis functions on simplex elements, as
well as frames based on piecewise polynomials conforming to the level set
function. The performance of the proposed method is compared to existing
adaptive multi-element generalized polynomial chaos methods
Motion Planning of Uncertain Ordinary Differential Equation Systems
This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems.
Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and under-actuated systems, prescribes deterministic actuator inputs which yield uncertain state trajectories. The inverse dynamics formulation is the dual to the forward dynamic, and is only applicable to fully-actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to under-actuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories which yield uncertain unactuated states and actuated inputs.
The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case-studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space
Multiscale Surrogate Modeling and Uncertainty Quantification for Periodic Composite Structures
Computational modeling of the structural behavior of continuous fiber
composite materials often takes into account the periodicity of the underlying
micro-structure. A well established method dealing with the structural behavior
of periodic micro-structures is the so- called Asymptotic Expansion
Homogenization (AEH). By considering a periodic perturbation of the material
displacement, scale bridging functions, also referred to as elastic correctors,
can be derived in order to connect the strains at the level of the
macro-structure with micro- structural strains. For complicated inhomogeneous
micro-structures, the derivation of such functions is usually performed by the
numerical solution of a PDE problem - typically with the Finite Element Method.
Moreover, when dealing with uncertain micro-structural geometry and material
parameters, there is considerable uncertainty introduced in the actual stresses
experienced by the materials. Due to the high computational cost of computing
the elastic correctors, the choice of a pure Monte-Carlo approach for dealing
with the inevitable material and geometric uncertainties is clearly
computationally intractable. This problem is even more pronounced when the
effect of damage in the micro-scale is considered, where re-evaluation of the
micro-structural representative volume element is necessary for every occurring
damage. The novelty in this paper is that a non-intrusive surrogate modeling
approach is employed with the purpose of directly bridging the macro-scale
behavior of the structure with the material behavior in the micro-scale,
therefore reducing the number of costly evaluations of corrector functions,
allowing for future developments on the incorporation of fatigue or static
damage in the analysis of composite structural components.Comment: Appeared in UNCECOMP 201
Computing derivative-based global sensitivity measures using polynomial chaos expansions
In the field of computer experiments sensitivity analysis aims at quantifying
the relative importance of each input parameter (or combinations thereof) of a
computational model with respect to the model output uncertainty. Variance
decomposition methods leading to the well-known Sobol' indices are recognized
as accurate techniques, at a rather high computational cost though. The use of
polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to
alleviate the computational burden though. However, when dealing with large
dimensional input vectors, it is good practice to first use screening methods
in order to discard unimportant variables. The {\em derivative-based global
sensitivity measures} (DGSM) have been developed recently in this respect. In
this paper we show how polynomial chaos expansions may be used to compute
analytically DGSMs as a mere post-processing. This requires the analytical
derivation of derivatives of the orthonormal polynomials which enter PC
expansions. The efficiency of the approach is illustrated on two well-known
benchmark problems in sensitivity analysis
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