Publications

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In order to impact physical mechanical system design decisions and realize the full promise of high-fidelity computational tools, simulation results must be integrated at the earliest stages of the design process. This is particularly challenging when dealing with uncertainty and optimizing for system-level performance metrics, as full-system models (often notoriously expensive and time-consuming to develop) are generally required to propagate uncertainties to system-level quantities of interest. Methods for propagating parameter and boundary condition uncertainty in networks of interconnected components hold promise for enabling design under uncertainty in real-world applications. These methods avoid the need for time consuming mesh generation of full-system geometries when changes are made to components or subassemblies. Additionally, they explicitly tie full-system model predictions to component/subassembly validation data which is valuable for qualification. These methods work by leveraging the fact that many engineered systems are inherently modular, being comprised of a hierarchy of components and subassemblies that are individually modified or replaced to define new system designs. By doing so, these methods enable rapid model development and the incorporation of uncertainty quantification earlier in the design process. The resulting formulation of the uncertainty propagation problem is iterative. We express the system model as a network of interconnected component models, which exchange solution information at component boundaries. We present a pair of approaches for propagating uncertainty in this type of decomposed system and provide implementations in the form of an open-source software library. We demonstrate these tools on a variety of applications and demonstrate the impact of problem-specific details on the performance and accuracy of the resulting UQ analysis. This work represents the most comprehensive investigation of these network uncertainty propagation methods to date.

This work summarizes the findings of a reduced order model (ROM) study performed using Sierra ROM module Pressio_Aria on Sandia National Laboratories' (SNL) Crash-Burn L2 milestone thermal model with pristine geometry. Comparisons are made to full order model (FOM) results for this same Crash-Burn model using Sierra multiphysics module Aria.

The design of thermal protection systems (TPS), including heat shields for reentry vehicles, rely more and more on computational simulation tools for design optimization and uncertainty quantification. Since high-fidelity simulations are computationally expensive for full vehicle geometries, analysts primarily use reduced-physics models instead. Recent work has shown that projection-based reduced-order models (ROMs) can provide accurate approximations of high-fidelity models at a lower computational cost. ROMs are preferable to alternative approximation approaches for high-consequence applications due to the presence of rigorous error bounds. The following paper extends our previous work on projection-based ROMs for ablative TPS by considering hyperreduction methods which yield further reductions in computational cost and demonstrating the approach for simulations of a three-dimensional flight vehicle. We compare the accuracy and potential performance of several different hyperreduction methods and mesh sampling strategies. This paper shows that with the correct implementation, hyperreduction can make ROMs up to 1-3 orders of magnitude faster than the full order model by evaluating the residual at only a small fraction of the mesh nodes.

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A projection-based reduced order model (pROM) methodology has been developed for transient heat transfer problems involving coupled conduction and enclosure radiation. The approach was demonstrated on two test problems of varying complexity. The reduced order models demonstrated substantial speedups (up to 185×) relative to the full order model with good accuracy (less than 3% L∞ error). An attractive feature of pROMs is that there is a natural error indicator for the ROM solution: the final residual norm at each time-step of the converged ROM solution. Using example test cases, we discuss how to interpret this error indicator to assess the accuracy of the ROM solution. The approach shows promise for many-query applications, such as uncertainty quantification and optimization. The reduced computational cost of the ROM relative to the full-order model (FOM) can enable the analysis of larger and more complex systems as well as the exploration of larger parameter spaces.

This work aims to advance computational methods for projection-based reduced-order models (ROMs) of linear time-invariant (LTI) dynamical systems. For such systems, current practice relies on ROM formulations expressing the state as a rank-1 tensor (i.e., a vector), leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern architectures. This weakness can be particularly limiting when tackling many-query studies, where one needs to run a large number of simulations. This work introduces a reformulation, called rank-2 Galerkin, of the Galerkin ROM for LTI dynamical systems which converts the nature of the ROM problem from memory bandwidth to compute bound. We present the details of the formulation and its implementation, and demonstrate its utility through numerical experiments using, as a test case, the simulation of elastic seismic shear waves in an axisymmetric domain. We quantify and analyze performance and scaling results for varying numbers of threads and problem sizes. Finally, we present an end-to-end demonstration of using the rank-2 Galerkin ROM for a Monte Carlo sampling study. We show that the rank-2 Galerkin ROM is one order of magnitude more efficient than the rank-1 Galerkin ROM (the current practice) and about 970 times more efficient than the full-order model, while maintaining accuracy in both the mean and statistics of the field.

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Melting and flowing of aluminum alloys is a challenging problem for computational codes. Unlike most common substances, the surface of an aluminum melt exhibits rapid oxidation and elemental migration, and like a bag filled with water can remain 2-dimensionally unruptured while the metal inside is flowing. Much of the historical work in this area focuses on friction welding and neglects the surface behavior due to the high stress of the application. We are concerned with low-stress melting applications, in which the bag behavior is more relevant. Adapting models and measurements from the literature, we have developed a formulation for the viscous behavior of the melt based on an abstraction of historical measurement, and a construct for the bag behavior. These models are implemented and demonstrated in a 3D level-set multi-phase solver package, SIERRA/Aria. A series of increasingly complex simulation scenarios are illustrated that help verify implementation of the models in conjunction with other required model components like convection, radiation, gravity, and surface interactions.

We propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying Kolmogorov n-width. However, these networks are restricted to data on structured meshes. Unstructured meshes are often required for performing analyses of real systems with complex geometry. Our custom graph convolution operators based on the available differential operators for a given spatial discretization effectively extend the application space of deep convolutional autoencoders to systems with arbitrarily complex geometry that are typically discretized using unstructured meshes. We propose sets of convolution operators based on the spatial derivative operators for the underlying spatial discretization, making the method particularly well suited to data arising from the solution of partial differential equations. We demonstrate the method using examples from heat transfer and fluid mechanics and show better than an order of magnitude improvement in accuracy over linear methods.

In order to impact design decisions and realize the full promise of high-fidelity computational tools, simulation results must be integrated at the earliest stages in the design process. This is particularly challenging when dealing with uncertainty and optimizing for system-level performance metrics as full-system models (often notoriously expensive and time-consuming to develop) are generally required to propagate uncertainties to system-level quantities of interest. Methods for propagating parameter and boundary condition uncertainty in networks of interconnected components hold promise for enabling design under uncertainty in real-world applications. These methods preclude the need for time consuming mesh generation of full-system geometries when changes are made to components or subassemblies. Additionally, they explicitly tie full-system model predictions to component/subassembly validation data which is valuable for qualification. This is accomplished by taking advantage of the fact that many engineered systems are inherently modular, being comprised of a hierarchy of components and subassemblies which are individually modified or replaced to define new system designs. We leverage this hierarchical structure to enable rapid model development and the incorporation of uncertainty quantification and rigorous sensitivity analysis earlier in the design process. The resulting formulation of the uncertainty propagation problem is iterative. We express the system model as a network of interconnected component models which exchange stochastic solution information at component boundaries. We utilize Jacobi iteration with Anderson acceleration to converge stochastic representations of system level quantities of interest through successive evaluations of component or subassembly forward problems. We publish our open-source tools for uncertainty propagation in networks remarking that these tools are extensible and can be used with any simulation tool (including arbitrary surrogate modeling tools) through the construction of a simple Python interface class. Additional interface classes for a variety of simulation tools are currently under active development. The performance of the uncertainty quantification method is determined by the number of iterations needed to achieve a desired level of accuracy. Performance of these networks for simple canonical systems from both a heat transfer and solid mechanics perspective are investigated; the models are examined with thermal and mechanical Dirichlet and Neumann type boundary conditions separately imposed and the impact of varying governing equations and boundary condition type on the performance of the networks is analyzed. The form of the boundary conditions is observed to have a large impact on the convergence rate with Neumann-type boundary conditions corresponding to significant performance degradation compared to the Dirichlet boundary conditions. Nonmonotonicity is observed in the solution convergence in some cases.

In this work, we revisit the classic problem of site percolation on a regular square lattice. In particular, we investigate the effect of quantization bias errors on percolation threshold predictions for large probability gradients and propose a mitigation strategy. We demonstrate through extensive computational experiments that the assumption of a linear relationship between probability gradient and percolation threshold used in previous investigations is invalid. Moreover, we demonstrate that, due to skewness in the distribution of occupation probabilities visited the average does not converge monotonically to the true percolation threshold. We identify several alternative metrics which do exhibit monotonic (albeit not linear) convergence and document their observed convergence rates.

We present a novel graph convolutional layer that is conceptually simple, fast, and provides high accuracy with reduced overfitting. Based on pseudo-differential operators, our layer operates on graphs with relative position information available for each pair of connected nodes. Our layer represents a generalization of parameterized differential operators (previously shown effective for shape correspondence, image segmentation, and dimensionality reduction tasks) to a larger class of graphs. We evaluate our method on a variety of supervised learning tasks, including 2D graph classification using the MNIST and CIFAR-100 datasets and 3D node correspondence using the FAUST dataset. We also introduce a superpixel graph version of the lesion classification task using the ISIC 2016 challenge dataset and evaluate our layer versus other state-of-the-art graph convolutional network architectures.The new layer outperforms multiple recent architectures on graph classification tasks using the MNIST and CIFAR-100 superpixel datasets. For the ISIC dataset, we outperform all other graph neural networks examined as well as all of the submissions to the original ISIC challenge despite the best of those models having more than 200 times as many parameters as our model.

Thermal protection system designers rely heavily on computational simulation tools for design optimization and uncertainty quantification. Because high-fidelity analysis tools are computationally expensive, analysts primarily use low-fidelity or surrogate models instead. In this work, we explore an alternative approach wherein projection-based reduced-order models (ROMs) are used to approximate the computationally infeasible high-fidelity model. ROMs are preferable to alternative approximation approaches for high-consequence applications due to the presence of rigorous error bounds. This work presents the first application of ROMs to ablation systems. In particular, we present results for Galerkin and least-squares Petrov-Galerkin ROMs of 1D and 2D ablation system models.

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