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DOE OSTI2020

Deep learning of free boundary and Stefan problems

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at https://github.com/PredictiveIntelligenceLab/DeepStefan.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Furthermore, adaptive griding results in more accurate computation of geometric quantities such as x-points associated with the model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Alluvial fan morphology: A self-similar free boundary problem description

In this work, we examine approximate geometrically self-similar solutions to a parabolic free boundary value problem applied to alluvial fan surface morphology and growth. Alluvial fans are fan- or cone-shaped sedimentary deposits caused by the rapid deposition of sediment from a canyon discharging onto a flatter plain. Longitudinal, topographic profiles of fans can be readily described by a seemingly time independent dimensionless profile (DeChant et al., 1999). However, because an alluvial fan can be expected to grow over time, it is clear that this “steady” profile is certainly time dependent and can be described using a space-time self-similar solution. In an experimental and theory-based study, Guerit et al. (2014) developed a self-similar (or as they describe it a self-affine) linear solution based upon an approximate first order small parameter expansion solution for a 1-d homogeneous nonlinear diffusion equation. Direct substitution of this result into a linear diffusion equation suggests that this first order expression may not fully satisfy the associated governing equation. In contrast, we develop a more complete solution based upon a modeled approximation for the axi-symmetric formulation such that the associated temporal behavior is consistent with a 1/3 time power-law as described by Reitz and Jerolmack (2014). The resulting expression is an exact solution to a linear heat equation. Furthermore, we emphasize that a small parameter is not inherent to the resulting profile result and is not included in our model development. Though developed using rather different approaches, the formal solution developed here is in good agreement with the simple polynomial described by DeChant et al. (1999) suggesting that this self-similar solution is a suitable time dependent representation of alluvial fan longitudinal profile form and improves on earlier work.

58 GEOSCIENCES↗

DOE OSTI2021

Surrogate approximation of the Grad–Shafranov free boundary problem via stochastic collocation on sparse grids

In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. Furthermore, the use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

A Parallel Cut-Cell Algorithm for the Free-Boundary Grad--Shafranov Problem

A parallel cut-cell algorithm is described to solve the free-boundary problem of the Grad--Shafranov equation. The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's acceleration for the resulting nonlinear problem, and a Cartesian grid embedded boundary method to handle the complex geometry. Here the algorithm is implemented in parallel using a standard domain-decomposition approach and a good parallel scaling is observed. Numerical results verify the accuracy and efficiency of the free-boundary Grad--Shafranov solver.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A weighted Shifted Boundary Method for free surface flow problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. Finally, we present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

A Virtual Element Method for the Full MHD system

In this manuscript we present a novel discretization for the incompressible MHD system. Our approach follows the framework of the Virtual Element Method and offers two main advantages. The method can be implemented in unstructured meshes making it highly versatile and capable of handling a wide array of problems involving interfaces, free-boundaries or adaptive refinements on the mesh. The second advantage involves the divergence of the magnetic field, our approach guarantees that it remains solenoidal. We include a theoretical proof of the condition on the magnetic field as well as energy estimates and a well-posedness study. The latter sheds light as to the stability properties of the method.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

An Adaptive Newton-Based Free-Boundary Grad–Shafranov Solver

Equilibria in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for the poloidal magnetic flux, commonly known as the Grad–Shafranov equation. It is challenging to develop a scalable and accurate free-boundary Grad–Shafranov solver, since it is a fully nonlinear optimization problem that simultaneously solves for the magnetic field coil current outside the plasma to control the plasma shape. In this work, we develop a Newton-based free-boundary Grad–Shafranov solver using adaptive finite elements and preconditioning strategies. The free-boundary interaction leads to the evaluation of a domain-dependent nonlinear form of which its contribution to the Jacobian matrix is achieved through shape calculus. The optimization problem aims to minimize the distance between the plasma boundary and specified control points while satisfying two nontrivial constraints, which correspond to the nonlinear finite element discretization of the Grad–Shafranov equation and a constraint on the total plasma current involving a nonlocal coupling term. The linear system is solved by a block factorization, and AMG is called for subblock elliptic operators. The unique contributions of this work include the treatment of a global constraint, preconditioning strategies, nonlocal reformulation, and the implementation of adaptive finite elements. Furthermore, it is found that the resulting Newton solver is robust, successfully reducing the nonlinear residual to 1e-6 and lower in a small handful of iterations while addressing the challenging case to find a Taylor state equilibrium where conventional Picard-based solvers fail to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Combined plasma–coil optimization algorithms

Combined plasma–coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi-free-boundary optimization, and the free-boundary (coil) optimization. These are described using the MRxMHD energy functional, the Biot–Savart integral, the coil-penalty functional and the virtual casing integral and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma–coil optimization algorithms, we emphasize the importance of the stability matrix.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Classifying Topology in Photonic Heterostructures with Gapless Environments

Photonic topological insulators exhibit bulk-boundary correspondence, which requires that boundary-localized states appear at the interface formed between topologically distinct insulating materials. However, many topological photonic devices share a boundary with free space, which raises a subtle but critical problem as free space is gapless for photons above the light line. Here, we use a local theory of topological materials to resolve bulk-boundary correspondence in heterostructures containing gapless materials and in radiative environments. In particular, we construct the heterostructure’s spectral localizer, a composite operator based on the system’s real-space description that provides a local marker for the system’s topology and a corresponding local measure of its topological protection; both quantities are independent of the material’s bulk band gap (or lack thereof). Moreover, we show that approximating radiative outcoupling as material absorption overestimates a heterostructure’s topological protection. Importantly, as the spectral localizer is applicable to systems in any physical dimension and in any discrete symmetry class (i.e., any Altland-Zirnbauer class), our results show how to calculate topological invariants, quantify topological protection, and locate topological boundary-localized resonances in topological materials that interface with gapless media in general.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2022

Variational encoder geostatistical analysis (VEGAS) with an application to large scale riverine bathymetry

Estimation of riverbed profiles, also known as bathymetry, plays a vital role in many applications, such as safe and efficient inland navigation, prediction of bank erosion, land subsidence, and flood risk management. The high cost and complex logistics of direct bathymetry surveys, i.e, depth imaging, have encouraged the use of indirect measurements such as surface flow velocities. However, estimating high-resolution bathymetry from indirect measurements is an inverse problem that can be computationally challenging. Here, we propose a reduced-order model (ROM) based approach that utilizes a variational autoencoder (VAE), a type of deep neural network with a narrow layer in the middle, to compress bathymetry and flow velocity information and accelerate bathymetry inverse problems from flow velocity measurements. In our application, the shallow-water equations (SWE) with appropriate boundary conditions (BCs), e.g., the discharge and/or the free surface elevation, constitute the forward problem, to predict flow velocity. Then, ROMs of the SWEs are constructed on a nonlinear manifold of low dimensionality through a variational encoder and the bathymetry inversion problem is derived on the low-dimensional latent space in a Hierarchical Bayesian setting. Further, the reformulation allows variational inference with a small number (e.g., $\mathscr{O}$ (100) of ROM runs and efficient uncertainty quantification. We have tested our inversion approach on a one-mile reach of the Savannah River, GA, USA. Once the neural network is trained (offline stage), the proposed technique can perform the inversion operation orders of magnitude faster than traditional inversion methods that are commonly based on linear projections, such as principal component analysis (PCA), or the principal component geostatistical approach (PCGA). Furthermore, tests show that the algorithm can estimate the bathymetry with good accuracy even with sparse flow velocity measurements.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

SIERRA Multimechanics Module: Aria User Manual - Version 4.56

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process ows via the incompressible Navier-Stokes equations specialized to a low Reynolds number (Re < 1) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newtons method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic h-adaptivity and dynamic load balancing are some of Arias more advanced capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

SIERRA Multimechanics Module: Aria User Manual - Version 4.58

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process flows via the incompressible Navier-Stokes equations specialized to a low Reynolds number ( %3C 1 ) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newtons method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic -adaptivity and dynamic load balancing are some of Arias more advanced capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

SIERRA Multimechanics Module: Aria User Manual (V.5.0)

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process %1Cows via the incompressible Navier-Stokes equations specialized to a low Reynolds number (Re < 1) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newtons method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic h-adaptivity and dynamic load balancing are some of Arias more advanced capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

SIERRA Multimechanics Module: Aria Thermal Theory Manual (V.5.0)

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process flows via the incompressible Navier-Stokes equations specialized to a low Reynolds number ( Re < 1) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newtons method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic -adaptivity and dynamic load balancing are some of Arias more advanced capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

SIERRA Multimechanics Module: Aria User Manual (V.5.4)

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process flows via the incompressible Navier-Stokes equations specialized to a low Reynolds number (Re < 1) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newton’s method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic ℎ-adaptivity and dynamic load balancing are some of Aria’s more advanced capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

SIERRA Multimechanics Module: Aria User Manual (Version.5.2)

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process flows via the incompressible Navier-Stokes equations specialized to a low Reynolds number (Re < 1) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newton’s method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic ℎ-adaptivity and dynamic load balancing are some of Aria’s more advanced capabilities.

97 MATHEMATICS AND COMPUTING↗