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Results for “Nonconforming methods”

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.

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At least 19 records
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DOE OSTI2018

The nonconforming virtual element method for eigenvalue problems

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L 2 -inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

The nonconforming Virtual Element Method for eigenvalue problems

In this work, we analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the $L^2$-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

The fully nonconforming virtual element method for biharmonic problems

Here in this paper, we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations

We present here the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov–Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection–diffusion–reaction problems in the convective-dominated regime. According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients. The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A meshless Galerkin method for non-local diffusion using localized kernel bases

Here, we introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Superconvergence analysis for nonlinear parabolic equation with EQ{sub 1}{sup rot} nonconforming finite element

EQ{sub 1}{sup rot} nonconforming finite element method (FEM) applied to a class of nonlinear parabolic equation is discussed. First, by use of two typical characters of this element (one is that the associated FE interpolation operator is identical to its traditional Ritz projection operator; the other is that the consistency error is of order O(h{sup 2}), one order higher than the interpolation error, when the exact solution of the problems belongs to H{sup 3}(Ω)), the supercloseness of order O(h{sup 2}) in broken H{sup 1}-norm for semi-discrete scheme is obtained without the boundedness of numerical solution in L{sup ∞}-norm through splitting the numerical solution into several parts. Moreover, we get the desired result with requirement of u,u{sub t}∈H{sup 3}(Ω) only. Secondly, a linearized Crank–Nicolson fully discrete scheme is proposed and the superclose property of order O(h{sup 2}+τ{sup 2}) is derived by constructing a suitable auxiliary problem. Finally, a numerical example is carried out to confirm our theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Large-Eddy Simulation over Complex Terrain Using an Improved Immersed Boundary Method in the Weather Research and Forecasting Model

Abstract The Weather Research and Forecasting (WRF) Model is increasingly being used for higher-resolution atmospheric simulations over complex terrain. With increased resolution, resolved terrain slopes become steeper, and the native terrain-following coordinates used in WRF result in numerical errors and instability. The immersed boundary method (IBM) uses a nonconformal grid with the terrain surface represented through interpolated forcing terms. Lundquist et al.’s WRF-IBM implementation eliminates the limitations of WRF’s terrain-following coordinate and was previously validated with a no-slip boundary condition for urban simulations and idealized terrain. This paper describes the implementation of a log-law boundary condition into WRF-IBM to extend its applicability to general atmospheric complex terrain simulations. The implementation of the improved WRF-IBM boundary condition is validated for neutral flow over flat terrain and the complex terrain cases of Askervein Hill, Scotland, and Bolund Hill, Denmark. First, comparisons are made to similarity theory and standard WRF results for the flat terrain case. Then, simulations of flow over the moderately sloped Askervein Hill are used to demonstrate agreement between the IBM and terrain-following WRF results, as well as agreement with observations. Finally, Bolund Hill simulations show that WRF-IBM can handle steep topography (standard WRF fails) and compares well to observations. Overall, the new WRF-IBM boundary condition shows improved performance, though the leeside representation of the flow can be potentially further improved.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Design, analysis and preliminary numerical results for the nonconforming VEM for parabolic problems

The virtual element method (VEM) is a numerical methodology for solving partial differential equations on unstructured polytopal meshes. In this work, we extend the nonconforming high-order formulation suitable to time-dependent diffusion problems with possibly nonlinear forcing terms. Coupling high-order virtual element formulations in space with time-marching schemes based on Backward Differention Formulas (BDFs) makes a fully discrete method with optimal order of convergence in the Method of Lines framework. We investigate and assess the performance of the method through theoretical analysis and a set of numerical experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

In fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finite-volume discretization for the multiphase flow equations coupled with a finite-element scheme for the mechanical equations. Fractures are treated as lower dimensional surfaces embedded in a background grid. Interactions are captured using the embedded discrete fracture model (EDFM) and the embedded finite element method (EFEM) for the flow and the mechanics, respectively. This nonconforming approach significantly alleviates meshing challenges. EDFM considers fractures as lower dimension finite volumes that exchange fluxes with the rock matrix cells. The EFEM method provides, instead, a local enrichment of the finite-element space inside each matrix cell cut by a fracture element. Both the use of piecewise constant and piecewise linear enrichments are investigated. They are also compared to an extended finite element approach. One key advantage of EFEM is the element-based nature of the enrichment, which reduces the geometric complexity of the implementation and leads to linear systems with advantageous properties. Synthetic numerical tests are presented to study the convergence and accuracy of the proposed method. It is also applied to a realistic scenario, involving a heterogeneous reservoir with a complex fracture distribution, to demonstrate its relevance for field applications.

58 GEOSCIENCES↗
DOE OSTI2019

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

In this work, we analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and we prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented. (An erratum is attached.)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A Condensed Constrained Nonconforming Mortar-Based Approach for Preconditioning Finite Element Discretization Problems

This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation that is based on a proposed modified version of the mortar method. The well-known mortar finite element discretization method is modified to admit a local structure, providing an element-by-element or subdomain-by-subdomain assembly property. This is achieved via the introduction of additional trace finite element spaces and degrees of freedom (unknowns) associated with the interfaces between adjacent elements or subdomains. The resulting nonconforming formulation and a reduced-via-static-condensation Schur complement form on the interfaces are used in the construction of auxiliary space preconditioners for a given conforming finite element discretization problem. Overall, the properties of these preconditioners are studied and their performance is illustrated on model second order scalar elliptic problems utilizing high order elements.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Multiscale design of nonlinear materials using a Eulerian shape optimization scheme

Motivated by recent advances in manufacturing, the design of materials is the focal point of interest in the material research community. One of the critical challenges in this field is finding optimal material microstructure for a desired macroscopic response. This work presents a computational method for the mesoscale-level design of particulate composites for an optimal macroscale-level response. The method relies on a custom shape optimization scheme to find the extrema of a nonlinear cost function subject to a set of constraints. Three key “modules” constitute the method: multiscale modeling, sensitivity analysis, and optimization. Multiscale modeling relies on a classical homogenization method and a nonlinear NURBS-based generalized finite element scheme to efficiently and accurately compute the structural response of particulate composites using a nonconformal discretization. A three-parameter isotropic damage law is used to model microstructure-level failure. An analytical sensitivity method is developed to compute the derivatives of the cost/constraint functions with respect to the design variables that control the microstructure's geometry. The derivation uncovers subtle but essential new terms contributing to the sensitivity of finite element shape functions and their spatial derivatives. Several structural problems are solved to demonstrate the applicability, performance, and accuracy of the method for the design of particulate composites with a desired macroscopic nonlinear stress-strain response.

42 ENGINEERING↗
DOE OSTI2021

Reanalysis of the top-quark pair hadroproduction and a precise determination of the top-quark pole mass at the LHC *

In this study, we calculate the $t\bar{t}$ pQCD production cross-section at the NNLO and determine the top-quark pole mass from recent measurements at the LHC at the center-of-mass energy $\sqrt{S}=13$ TeV to a high precision by applying the principle of maximum conformality (PMC). The PMC provides a systematic method that rigorously eliminates QCD renormalization scale ambiguities by summing the nonconformal $\beta$ contributions into the QCD coupling constant. The PMC predictions satisfy the requirements of renormalization group invariance, including renormalization scheme independence, and the PMC scales accurately reflect the virtuality of the underlying production subprocesses. By using the PMC, an improved prediction for the $t\bar{t}$ production cross-section is obtained without scale ambiguities, which in turn provides a precise value for the top-quark pole mass. Moreover, the prediction of PMC calculations that the magnitudes of higher-order PMC predictions are well within the error bars predicted from the known lower-order has been demonstrated for the top-quark pair production. The resulting determination of the top-quark pole mass, $m_t^{\rm{pole}}=172.5\pm1.4$ GeV, from the LHC measurement at $\sqrt{S}=13$ TeV agrees with the current world average cited by the Particle Data Group (PDG). The PMC prediction provides an important high-precision test of the consistency of pQCD and the SM at $\sqrt{S}=13$ TeV with previous LHC measurements at lower CM energies.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Self-consistent analysis for the η c → γ γ process

The next-to-next-to-leading-order (NNLO) perturbative QCD (pQCD) predictions for both the decay width and the transition form factor in the η c → γ γ process, based on nonrelativistic QCD (NRQCD), deviate from precise experimental measurements. These significant discrepancies have cast doubt on the applicability of NRQCD to charmonium processes. In this paper, we analyze the η c → γ γ process by applying the principle of maximum conformality (PMC), a systematic method for eliminating renormalization scheme-and-scale ambiguities. The PMC renormalization scales are determined by absorbing the nonconformal β -terms that govern the behavior of the QCD running coupling via the renormalization group equation. We obtain the PMC scale Q ⋆ = 4.49 m c for the η c → γ γ decay width. Even after using the PMC method, the convergence of the pQCD series is still poor, which indicates the importance of uncalculated next-to-next-to-next-to-leading-order and higher-order terms. The resulting value for Γ η c → γ γ is in agreement with the Particle Data Group’s reported value of Γ η c → γ γ = 5.1 ± 0.4 keV within the bounds of uncertainties. Moreover, the transition form factor obtained using the PMC is also in good agreement with precise experimental measurements. The application of the PMC suggests a potential resolution to η c → γ γ puzzle and supports the applicability of NRQCD to charmonium processes. Published by the American Physical Society 2025

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Auxiliary Space Preconditioning of Finite Element Equations Using a Nonconforming Interior Penalty Reformulation and Static Condensation

We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced) version of it, are utilized in a number of auxiliary space preconditioners for the original conforming finite element discretization problem. Furthermore, these preconditioners are analyzed on the fine scale, and their performance is illustrated on model second order scalar elliptic problems discretized with high order elements.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A cell-centered AMR-ALE framework for 3D multi-material hydrodynamics. Part II: linesweep ALE rezoning for nonconformal block-structured AMR meshes

The simulation of flows presenting contact discontinuities, vorticity, and large variations in spatial scales can be performed in a framework coupling Arbitrary Lagrangian Eulerian (ALE) algorithms and Adaptive Mesh Refinement (AMR). This coupling requires adaptation of ALE rezoning techniques to meshes containing nonconformal nodes arising from both the AMR topology and the junction of mesh blocks. Here, in this paper, we present an ALE rezoning strategy that is compatible with such meshes, and that can also act as a disentangling algorithm. Emphasis is put on an algorithm that respects intrinsic Lagrangian mesh properties in order to preserve accuracy around discontinuities. To that end, we adapt the weighted linesweep algorithm to nonconformal block-structured AMR meshes. Then, we present control parameters introduced in the method for it to be applicable in practical situations. Notably, the method is coupled to a specific metric optimization in order to palliate some shortcomings of the linesweep method. Finally, numerical test cases are presented that feature the capabilities of the ALE-AMR algorithm for flows that present discontinuities, vorticity, and a variety of scales. Notably, we show that our ALE-AMR algorithm gives results at least similar to Euler-AMR, but provides better accuracy in cases where discontinuities are involved, thanks to a method that respects the Lagrangian features of the mesh. Additionally, it enables Euler-AMR-like computations on domains with temporally varying domain boundaries.

Adaptive mesh refinement↗