Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Elliptic problems”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2013

SUPERMASSIVE BLACK HOLE BINARY EVOLUTION IN AXISYMMETRIC GALAXIES: THE FINAL PARSEC PROBLEM IS NOT A PROBLEM

During a galaxy merger, the supermassive black hole (SMBH) in each galaxy is thought to sink to the center of the potential and form an SMBH binary; this binary can eject stars via three-body scattering, bringing the SMBHs ever closer. In a static spherical galaxy model, the binary stalls at a separation of about a parsec after ejecting all the stars in its loss cone-this is the well-known final parsec problem. Earlier work has shown that the centrophilic orbits in triaxial galaxy models are key in refilling the loss cone at a high enough rate to prevent the black holes from stalling. However, the evolution of binary SMBHs has never been explored in axisymmetric galaxies, so it is not clear if the final parsec problem persists in these systems. Here we use a suite of direct N-body simulations to follow SMBH binary evolution in galaxy models with a range of ellipticity. For the first time, we show that mere axisymmetry can solve the final parsec problem; we find the SMBH evolution is independent of N for an axis ratio of c/a = 0.8, and that the SMBH binary separation reaches the gravitational radiation regime for c/a = 0.75.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2018

Hybrid Finite Element--Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver

Here, a numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $\Omega$ on the extruded domain $\mathcal{C}=\Omega\times[0,\infty)$ following. The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme. Finite elements are used in the direction parallel to the problem domain $\Omega$, and an appropriate spectral method is used in the extruded direction. The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over $\Omega$. We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues. We further present a strategy for choosing approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem. The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient. Numerical examples in two and three dimensions suggest that the approach is quasi-optimal in terms of complexity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

PDE-constrained Optimization under Uncertainty

Uncertainty is pervasive in all science and engineering applications. Incorporating uncertainty in physical models is therefore both natural and vital. In doing so, we often arrive at parametric systems of partial differential equations (PDEs). When passing from simulation to optimization, we obtain (typically nonconvex) infinite-dimensional optimization problems that, upon discretization, result in extremely large-scale nonlinear programs. For example, consider a linear elliptic PDE on a two- dimensional domain with a single random coeffcient. If we sampled the random input with 10,000 realizations of the coeffcient, the resulting optimization problem would have 10,000 PDE constraints. Furthermore, discretizing each PDE with piecewise linear finite elements on a 100X100 uni- form quadrilateral mesh results in 100,000,000 degrees of freedom. As a result, the critical components for ensuring mesh-independent performance of numerical optimization methods in the deterministic setting, for example, solution regularity and generalized differentiability, are even more critical in the stochastic setting.

42 ENGINEERING↗
DOE OSTI2016

An added-mass partition algorithm for fluid–structure interactions of compressible fluids and nonlinear solids

We describe an added-mass partitioned (AMP) algorithm for solving fluid–structure interaction (FSI) problems involving inviscid compressible fluids interacting with nonlinear solids that undergo large rotations and displacements. The computational approach is a mixed Eulerian–Lagrangian scheme that makes use of deforming composite grids (DCG) to treat large changes in the geometry in an accurate, flexible, and robust manner. The current work extends the AMP algorithm developed in Banks et al. [1] for linearly elasticity to the case of nonlinear solids. To ensure stability for the case of light solids, the new AMP algorithm embeds an approximate solution of a nonlinear fluid–solid Riemann (FSR) problem into the interface treatment. The solution to the FSR problem is derived and shown to be of a similar form to that derived for linear solids: the state on the interface being fundamentally an impedance-weighted average of the fluid and solid states. Numerical simulations demonstrate that the AMP algorithm is stable even for light solids when added-mass effects are large. The accuracy and stability of the AMP scheme is verified by comparison to an exact solution using the method of analytical solutions and to a semi-analytical solution that is obtained for a rotating solid disk immersed in a fluid. The scheme is applied to the simulation of a planar shock impacting a light elliptical-shaped solid, and comparisons are made between solutions of the FSI problem for a neo-Hookean solid, a linearly elastic solid, and a rigid solid. The ability of the approach to handle large deformations is demonstrated for a problem of a high-speed flow past a light, thin, and flexible solid beam.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension

Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order Runge–Kutta method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified Harten–Lax–van Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gas–liquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Here, we propose an extension of the embedded boundary method known as “shifted boundary method” to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions

Highlights: Complete classification of near-neighbors CS models with a Jastrow-like ground state. Examples of such models with rational, trigonometric and hyperbolic interactions. Construction of a new elliptic model which yields these examples in suitable limits. -- Abstract: We completely solve the problem of classifying all one-dimensional quantum potentials with nearest- and next-to-nearest-neighbors interactions whose ground state is Jastrow-like, i.e., of Jastrow type but depending only on differences of consecutive particles. In particular, we show that these models must necessarily contain a three-body interaction term, as was the case with all previously known examples. We discuss several particular instances of the general solution, including a new hyperbolic potential and a model with elliptic interactions which reduces to the known rational and trigonometric ones in appropriate limits.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Time-explicit Darwin PIC algorithm

A new approach to Darwin particle-in-cell plasma simulation is described. Using a finite-element approach, the vector potential is assured to be exactly solenoidal. This allows writing the system of multi-species particles as an action-at-a-distance Hamiltonian system. Applying recently-developed explicit symplectic methods to this gives a time-explicit algorithm with all desired properties. Elliptic systems for the electrostatic and magneto-static portions of the problem are inverted efficiently by an algebraic multi-grid algorithm. The algorithm is implemented in a two-dimensional Cartesian-geometry code and tested by application to Weibel instability and to Alfvén wave dynamics. Results show the effectiveness of this approach. Extensions to arbitrary meshes and to a partially time-implicit scheme are briefly discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

A posteriori snapshot location for POD in optimal control of linear parabolic equations

In this paper we study the approximation of an optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. We propose to determine the time instances (snapshot locations) by an a posteriori error control concept. The proposed method is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and show the effectiveness of the method in comparison to existing approaches.

Mathematics↗
DOE OSTI2020

Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations

We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve forward, inverse, and mixed stochastic problems in a unified manner based on a limited number of scattered measurements. Unlike standard GANs relying solely on data for training, here we encoded into the architecture of GANs the governing physical laws in the form of stochastic differential equations (SDEs) using automatic differentiation. In particular, we applied Wasserstein GANs with gradient penalty (WGAN-GP) for its enhanced stability compared to vanilla GANs. We first tested WGAN-GP in approximating Gaussian processes of different correlation lengths based on data realizations collected from simultaneous reads at sparsely placed sensors. We obtained good approximation of the generated stochastic processes to the target ones even if there is a mismatch between the input noise dimensionality and the effective dimensionality of the target stochastic processes. We also studied the overfitting issue for both the discriminator and the generator, and we found that overfitting occurs also in the generator in addition to the discriminator as previously reported. Subsequently, we considered the solution of elliptic SDEs requiring approximations of three stochastic processes, namely the solution, the forcing, and the diffusion coefficient. Here again, we assumed data collected from simultaneous reads at a limited number of sensors for the multiple stochastic processes. Furthermore, three generators were used for the PI-GANs: two of them were feed forward deep neural networks (DNNs), while the other one was the neural network induced by the SDE. For the case where we have one group of data, we employed one feed forward DNN as the discriminator, while for the case of multiple groups of data we employed multiple discriminators in PI-GANs. We solved forward, inverse, and mixed problems without changing the framework of PI-GANs, obtaining both the means and the standard deviations of the stochastic solution and the diffusion coefficient in good agreement with benchmarks. In this work, we have demonstrated the effectiveness of PI-GANs in solving SDEs for about 120 dimensions. In principle, PI-GANs could tackle very high dimensional problems given more sensor data with low-polynomial growth in computational cost.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Dynamical friction in cuspy galaxies

In this paper, we treat the problem of the dynamical friction decay of a massive object moving in an elliptical galaxy with a cuspidal inner distribution of the mass density. We present results obtained by both self-consistent, direct summation, N-body simulations, as well as by a new semi-analytical treatment of dynamical friction valid in such cuspy central regions of galaxies. A comparison of these results indicates that the proposed semi-analytical approximation is the only reliable one in cuspy galactic central regions, where the standard Chandrasekhar's local approximation fails and also gives estimates of decay times that are correct at 1% with respect to those given by N-body simulations. The efficiency of dynamical friction in cuspy galaxies is found definitively higher than in core galaxies, especially on more radially elongated satellite orbits. As another relevant result, we find a proportionality of the dynamical friction decay time to the –0.67 power of the satellite mass, M, shallower than the standardly adopted M {sup –1} dependence.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

On non-autonomous dynamical systems

In usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the Poisson-Lie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

αAMG based on Weighted Matching for Systems of Elliptic PDEs Arising From Displacement and Mixed Methods

Adaptive Algebraic Multigrid (or Multilevel) Methods (αAMG) are introduced to improve robustness and efficiency of classical algebraic multigrid methods in dealing with problems where no a-priori knowledge or assumptions on the near-null kernel of the underlined matrix are available. Recently we proposed an adaptive (bootstrap) AMG method, αAMG, aimed to obtain a composite solver with a desired convergence rate. Each new multigrid component relies on a current (general) smooth vector and exploits pairwise aggregation based on weighted matching in a matrix graph to define a new automatic, general-purpose coarsening process, which we refer to as “the compatible weighted matching”. In this work, we present results that broaden the applicability of our method to different finite element discretizations of elliptic PDEs. In particular, we consider systems arising from displacement methods in linear elasticity problems and saddle-point systems that appear in the application of the mixed method to Darcy problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

A hyperbolic-equation system approach for magnetized electron fluids in quasi-neutral plasmas

A new approach using a hyperbolic-equation system (HES) is proposed to solve for the electron fluids in quasi-neutral plasmas. The HES approach avoids treatments of cross-diffusion terms which cause numerical instabilities in conventional approaches using an elliptic equation (EE). A test calculation reveals that the HES approach can robustly solve problems of strong magnetic confinement by using an upwind method. The computation time of the HES approach is compared with that of the EE approach in terms of the size of the problem and the strength of magnetic confinement. The results indicate that the HES approach can be used to solve problems in a simple structured mesh without increasing computational time compared to the EE approach and that it features fast convergence in conditions of strong magnetic confinement.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Smoothed aggregation adaptive spectral element-based algebraic multigrid

SAAMGE provides parallel methods for building multilevel hierarchies and solvers that can be used for elliptic equations with highly heterogeneous coefficients. Additionally, hierarchy adaptation is implemented allowing solving multiple problems with close coefficients without rebuilding the hierarchy.

Source record↗
DOE OSTI2024

Asymptotic quasisymmetric high-beta three-dimensional magnetohydrodynamic equilibria near axisymmetry

Quasisymmetry (QS), a hidden symmetry of the magnetic field strength, is known to support nested flux surfaces and provide superior particle confinement in stellarators. In this work, we study the ideal magnetohydrodynamic (MHD) equilibrium and stability of high-beta plasma in a large-aspect-ratio stellarator. In particular, we show that the lowest-order description of a near-axisymmetric equilibrium vastly simplifies the problem of three-dimensional quasisymmetric MHD equilibria, which can be reduced to a standard elliptic Grad–Shafranov equation for the flux function. We show that any large-aspect-ratio tokamak, deformed periodically in the vertical direction, is a stellarator with approximate volumetric QS. We discuss exact analytical solutions and numerical benchmarks. Finally, we discuss the ideal ballooning and interchange stability of some of our equilibrium configurations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Modeling Simulation and Optimization of a Pressure Relief Feature in a Hermetically Sealed Volume Exposed to Abnormal Thermal Environments

This project investigates the behavior of a venting feature on a hermetically sealed volume under abnormal thermal environments. A pressure profile is used to simulate internal pressure build up due to thermal decomposition of foam caused by temperatures approaching 800K. For the purposes of his scoping study a small can composed of 304L stainless steel is used to emulate the problem at hand and simplify the model. The venting feature can be described as an elliptical thinned area with X-like scoring marks on the circumferential surface of the can where the highest stresses are located without the feature. The feature variables include elliptical size, depth, and quantity to optimize the venting feature geometry to vent at a desired pressure range. Cubit, SIERRA: Solid Mechanics, and Ensight modeling programs are used to simulate and analyze several iterations of this feature to determine the most optimal geometry. The model indicates through the simulations that the feature does in fact vent in a predictable manner and can be used in a variety of pressure vessel applications where uncontrolled over pressurizations are undesirable. Future work includes model validation tests, looking at different geometries, and using a more accurate failure criterion.

42 ENGINEERING↗
DOE OSTI2016

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddle-point system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver. Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.

97 MATHEMATICS AND COMPUTING↗