Local Discretization Research Articles

A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

This paper is concerned with the numerical solution of the flow problem in a fractured porous medium where the fracture is treated as a lower dimensional object embedded in the rock matrix. We consider a space-time mixed variational formulation of such a reduced fracture model with mixed finite element approximations in space and discontinuous Galerkin discretization in time. Different spatial and temporal grids are used in the subdomains and in the fracture to adapt to the heterogeneity of the problem. Analysis of the numerical scheme, including well-posedness of the discrete problem, stability and a priori error estimates, is presented. Using substructuring techniques, the coupled subdomain and fracture system is reduced to a space-time interface problem which is solved iteratively by GMRES. Each GMRES iteration involves solution of time-dependent problems in the subdomains using the method of lines with local spatial and temporal discretizations. The convergence of GMRES is proved by using the field-of-values analysis and the properties of the discrete space-time interface operator. Numerical experiments are carried out to illustrate the performance of the proposed iterative algorithm and the accuracy of the numerical solution.

Extension of the local domain-free discretization method to large eddy simulation of compressible flows

Most of the flow problems encountered in practical engineering are wall-bounded turbulent flows at high Reynolds numbers. Wall-modeled large eddy simulation (WMLES) is one of the most viable approaches for predicting these realistic flows. Immersed boundary (IB) approach is an efficient computational technique to solve flow problems involving complex and/or moving geometries. This work extends a sharp-interface IB method, named the local domain-free discretion (DFD), to WMLES of compressible flows at high Reynolds numbers. An equilibrium wall model based on solving the simplified compressible turbulent boundary layer equations is utilized to alleviate the requirement of high near-wall mesh resolution. In conjunction with the approximate boundary conditions prescribed by the modeled wall shear stress and wall heat flux, the tangential velocity and temperature at an exterior dependent node are evaluated. Then, the closure of the discrete form of governing equations at an interior node in the immediate vicinity of the immersed wall is accomplished. A simple non-equilibrium correction of the wall shear stress provided by the equilibrium wall model is introduced explicitly. The WMLES/DFD method is applied to a supersonic zero-pressure-gradient turbulent boundary layer flow, a shock wave/flat-plate boundary layer interaction, a supersonic compression ramp flow and high-speed turbulent Couette flows with various thermal boundary conditions. The influence of grid resolution is investigated in the simulation of zero-pressure-gradient turbulent boundary layer flow. By comparing the computed results with the referenced experimental data and/or numerical results, the accuracy and ability of the WMLES/DFD method to simulate compressible turbulent flows are verified.

Domain decomposition with local time discretization for the nonlinear Stokes–Biot system

This work presents a domain decomposition method for the fluid-poroelastic structure interaction (FPSI) system, which utilizes local time integration for subproblems. To derive the domain decomposition scheme, we introduce a Lagrange multiplier and define time-dependent Steklov–Poincaré-type operators based on the interface conditions. These operators are employed to transform the coupled system into an evolutionary nonlinear interface problem, which is then solved using an iterative algorithm. This approach provides the flexibility to use different time discretization schemes and step sizes in subdomains, making it an efficient method for simulating multiphysics systems. We present numerical tests for both non-physical and physical problems to demonstrate the accuracy and efficiency of this method.

Helical Luttinger Liquid on a Space-Time Lattice.

The Luttinger model is a paradigm for the breakdown due to interactions of the Fermi liquid description of one-dimensional massless Dirac fermions. Attempts to discretize the model on a one-dimensional lattice have failed to reproduce the established bosonization results because of the fermion-doubling obstruction: a local and symmetry-preserving discretization of the Hamiltonian introduces a spurious second species of low-energy excitations, while a nonlocal discretization opens a single-particle gap at the Dirac point. Here, we show how to work around this obstruction by discretizing both space and time to obtain a local Lagrangian for a helical Luttinger liquid with Hubbard interaction. The approach enables quantum MonteCarlo simulations that preserve the topological protection of an unpaired Dirac cone.

A general pressure equation based method for incompressible two‐phase flows

SummaryWe present a fully‐explicit, iteration‐free, weakly‐compressible method to simulate immiscible incompressible two‐phase flows. To update pressure, we circumvent the computationally expensive Poisson equation and use the general pressure equation which is solved explicitly. In addition, a less diffusive algebraic volume‐of‐fluid approach is used as the interface capturing technique and in order to facilitate improved parallel computing scalability, the technique is discretised temporally using the operator‐split methodology. Our method is fully‐explicit and stable with simple local spatial discretization, and hence, it is easy to implement. Several two‐ and three‐dimensional canonical two‐phase flows are simulated. The qualitative and quantitative results prove that our method is capable of accurately handling problems involving a range of density and viscosity ratios and surface tension effects.

A wall-modeled immersed boundary/large eddy simulation method and its application to simulating heart valve flows

In this work, a wall-modeled immersed boundary (IB)/large eddy simulation (LES) method is extended to the simulation of moving-boundary flows. The used non-equilibrium algebraic wall model is based on an assumed velocity profile, the coefficients of which are determined from physical constraints provided by the full turbulent-boundary-layer equations. To implement the wall model in an IB method named the local domain-free discretization (DFD) method, a local coordinate system fixed on the moving body is introduced. Thus, wall modeling is transformed into a local two-dimensional problem and the complexity of implementation of the wall model is reduced. In the present LES-DFD method, the tangential velocity at an exterior dependent node is determined via wall shear stress prescribed by the wall model. To reduce computational cost for simulating an internal flow with moving boundaries, the stationary boundaries are handled by the body-fitted-grid method and the moving boundaries by the local DFD method. There is no need of an auxiliary grid for solving the non-equilibrium algebraic wall model. Therefore, the inbuilt advantage of an IB method can be retained when simulating moving-boundary problems, and the economy of equilibrium wall models can also be preserved. The present method is applied to simulating the pulsatile flows through a bileaflet mechanical heart valve implanted in a model aorta. The predicted results show an acceptable agreement with the referenced experimental measurements or numerical results at much higher resolution and the applicability of the non-equilibrium wall model to LES of complex moving-boundary flows is verified.

Topologically protected Casimir effect for lattice fermions

The electromagnetic Casimir effect has a fermionic counterpart in topological insulators: Zero-point fluctuations of a massless Dirac fermion field mediate a force between magnetic scatterers. The Casimir force is insensitive to disorder that preserves the topological protection of an unpaired Dirac cone. The protection may be broken if the Dirac equation is discretized, and an exponential suppression of the Casimir effect will result if a gap opens at the Dirac point. Here, we show how this lattice artifact may be avoided, by applying a recently developed local discretization of the Euclidean action that does not suffer from the fermion-doubling obstruction of local discretizations of the Hamiltonian. Published by the American Physical Society 2024

Local discontinuous Galerkin for the functional renormalisation group

We apply a Local Discontinuous Galerkin discretisation to flow equations of the O(N)-model in the Local Potential Approximation. The improved stability is directly observed by solving the flow equation for various N and space-time dimensions d. A particular focus of this work is the numerical discretisation and its implementation. It is realised as a module within the high performance PDE framework DUNE. A preliminary version of the module is available on GitHub, but it is not submitted for publication as it is too early to be archived.

A GFDM approach based on the finite pointset method for two-dimensional piezoelectric problems

In this article a novel Generalized Finite Difference Method (GFDM) derived from the so-called Finite Pointset Method (FPM) is presented and discussed for the first time to solve two-dimensional piezoelectric structures. In this approach, the approximation of the field variables depends on both the governing equations and the local problem discretization, and it incorporates the minimization of the errors of the governing equations being solved, which directly leads to its numerical solution. This truly meshfree formulation is applied in order to solve a static piezoelectric problem that consists of the coupled partial differential equations for the deformations and the electric field, in terms of the displacements and the scalar electric potential. The numerical results of some two-dimensional benchmark problems using the proposed approach are reported and compared with analytical solutions, if available, or with reference numerical solutions, in order to demonstrate its accuracy and suitability for this kind of problems. These numerical results suggest that the proposed GFDM formulation is feasible and promising for solving two-dimensional piezoelectric problems.

Implementation of a dynamic slip wall model for large eddy simulation in the local domain-free discretization method

Wall-modeling techniques are frequently applied in the extension of immersed boundary methods to large eddy simulation (LES) of turbulent flows. In this paper, a dynamic slip wall model is incorporated into an immersed boundary method named the local domain-free discretization (DFD) method for implicitly filtered LES, which is based on a Robin-type boundary condition and free of a priori specified tunable coefficients and also does not rely on any assumptions about the boundary layer. The present study provides a detailed description of the implementation of the dynamic slip model in the DFD framework, including the evaluation of the velocity components at an exterior dependent node via the Robin-type boundary condition and the dynamic procedure for calculating the slip length. The non-zero slip velocity in the normal direction is imposed in a new way to satisfy on average the non-penetration of the curved wall and maintain the global mass conservation in the implicitly filtered LES. To validate the present wall-modeling LES/DFD method, a series of turbulent channel flows at various Reynolds numbers, the flow over periodic hills, and the high-Reynolds-number flow past a National Advisory Committee for Aeronautics (NACA) 4412 airfoil at near-stall conditions are simulated, and the predicted results agree well with the referenced experimental and numerical results.

Mitigated barren plateaus in the time-nonlocal optimization of analog quantum-algorithm protocols

Quantum machine learning has emerged as a promising utilization of near-term quantum computation devices. However, algorithmic classes such as variational quantum algorithms have been shown to suffer from barren plateaus due to vanishing gradients in their parameters spaces. We present an approach to quantum algorithm optimization that is based on trainable Fourier coefficients of Hamiltonian system parameters. Our is exclusive to the extension of discrete quantum variational algorithms to analog quantum optimal control schemes and is nonlocal in time. We demonstrate the viability of our on the objectives of compiling the quantum Fourier transform and preparing ground states of random problem Hamiltonians. In comparison to the temporally local discretization in quantum optimal control and parametrized circuits, our exhibits faster and more consistent convergence. We uniformly sample objective gradients across the parameter space and find that in our the variance decays at a nonexponential rate with the number of qubits, while it decays at an exponential rate in the temporally local benchmark . This indicates the mitigation of barren plateaus in our . We propose our as a viable candidate for near-term quantum machine learning. Published by the American Physical Society 2024

Discretization Error Estimation Using the Unsteady Error Transport Equations

Abstract Computational Fluid Dynamics (CFD) has gained significant utility in the analysis of diverse fluid flow scenarios, thanks to advances in computational power. Accurate error estimation techniques are crucial to ensure the reliability of CFD simulations, as errors can lead to misleading conclusions. This study focuses on the estimation of discretization errors in time-dependent simulations, building upon prior work addressing steady-state problems Wang et al. (2020, “Error Transport Equation Implementation in the Sensei CFD Code,” AIAA Paper No. 2020-1047). In this research, we employ unsteady error transport equations (ETE) to generate localized discretization error estimates within the framework of the finite volume CFD code SENSEI. For steady-state problems, the ETE only need to be solved once after the solution has converged, whereas the unsteady ETE need to be co-advanced with the primal solve. To enhance efficiency, we adopt a one-sided temporal stencil and develop a modified iterative correction process tailored to the unsteady ETE. The time-marching schemes utilized encompass second-order accurate singly diagonally implicit Runge–Kutta (SDIRK) and second-order backward differentiation formula (BDF2), both being implicit. To rigorously assess the accuracy of our error estimates, all test cases feature known analytical solutions, facilitating order-of-accuracy evaluations. Two test cases are considered: the 2D convected vortex for inviscid flow and a cross-term sinusoidal (CTS) manufactured solution for viscous flow. Results indicate higher-order convergence rates for the 2D convected vortex test case even when iterative correction is not applied, with similar observations in the CTS case, albeit not at the finest grid levels. Although the current implementation of iterative correction exhibits lower stability compared to the primal solve, it generally enhances the discretization error estimate. Notably, after iterative correction, the discretization error estimate for the unsteady ETE achieves higher-order accuracy across all grid levels in the 2D CTS manufactured solution.

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

A whole discretization method for milling stability prediction considering the discrete vibration velocities

The discretization methods have been widely used for chatter stability prediction and can be divided into two categories: semi-discretization methods (SDMs) by discretizing the time-delayed displacements, and full-discretization methods (FDMs) by discretizing both the present and time-delayed displacements. In this paper, a whole discretization method is proposed to achieve a faster and more accurate prediction for the milling stability. In addition to the discrete vibration displacements, the discrete vibration velocities are included in the solution by rebuilding the integrated matrix. The prediction error caused by the separate discretization of the cutting force coefficient matrix and vibration state variables can be avoided. The milling stability is determined with the discrete map of the state based on the Floquet theory. The proposed model is compared with the zeroth-order SDM and the first-order FDM which also approximate the solution with two neighboring discrete state variables. It is shown that the local discretization error based on the proposed method achieves o((Δt)5), which is three orders higher compared to the zeroth-order SDM and the first-order FDM. In addition, when the same prediction errors are required, the computational time of the proposed method is 11–14% of the zeroth-order SDM, and 24–27% of the first-order FDM. The results prove that considering vibration velocity can increase the efficiency and accuracy of the chatter stability prediction compared to existing SDM and FDM algorithms.

Digital twin of an apparatus for combined thermoelectric measurements

A high thermoelectric (TE) figure of merit zT of materials enables a high energy conversion efficiency. The quantity zT is defined by the Seebeck coefficient (S), the electric (σ) and thermal (κ) conductivity, and the absolute temperature (T). In this paper, we report on a computational model of the Combined ThermoElectric Measurement (CTEM) apparatus, which is a simultaneous characterization method capable of measuring the full set of above-mentioned thermoelectric transport properties between −190 and 600 °C. Currently, the measurement results show deviations due to unidentified error sources. As a solution approach of identifying possible error sources, a digital twin of the CTEM was developed. The computational thermal-electrical circuit model mainly consists of thermal sieving chains representing the relevant sample holder components, in particular two metallic blocks and a TE sample. For a computational consistency check of the measuring principles, ideal conditions are assumed, while no potential error sources are implemented, yet. Here, we present the measurement principles and procedures of creating the computational model of the CTEM. After studies on local discretization, the computational model undergoes a consistency check for model validation. The deviations between input parameters and simulated results of the three mentioned thermoelectric properties have been found negligibly small (≪1%) for ideal measurement conditions. This agreement certifies a realistic representation of the behavior of the sample holder by the digital twin with a satisfying reproduction of ideal measurement conditions by simplifying assumptions and the applicability of underlying measurement principles and evaluation protocols.

Modelling edge effects at the interface in bonded joints using gradient functions in the mechanical properties of the adhesive: Application of the method to the Arcan test loaded in tension and shear

The design of bonded joints requires studies of stress concentrations due to edge effects. For complex joint configurations, the finite element method can be quite costly. The objective is to develop a fast and reliable numerical design tool for bonded assemblies. Therefore, an approach to design bonded assemblies is presented which aims to meet the needs of a design office, particularly in terms of calculation costs. The latter consists in reproducing the edge effects with a single element through the joint thickness using a modulus function. The concept was tested on Arcan specimen with two loading cases: tension (γ = 0°) and shear (γ = 90°). A 2D model under the elastic assumption is developed to describe the edge effects of the joint using Abaqus subroutines (UMat). The approach is set up to solve this problem in the form of two blocks. First, mesh refinement studies for bonded specimens loaded in tension were performed, within a good level of accuracy, on the free edge with the use of local discretization error estimation. After that, the effects of local geometry and modulus ratio are investigated. Afterwards, the von Mises stress at the interface level of the adhesive joint was used to identify a modulus function to describe the behavior of a joint with straight edge geometry for tension and shear loadings. The proposed approach has improved the performance of the model. Actually, the calculation is practically three times faster than for the conventional model.

Stability and Error Estimates of a Novel Spectral Deferred Correction Time-Marching with Local Discontinuous Galerkin Methods for Parabolic Equations

Abstract In this paper, we discuss the stability and error estimates of the fully discrete schemes for parabolic equations, in which local discontinuous Galerkin methods with generalized alternating numerical fluxes and a novel spectral deferred correction method based on second-order time integration methods are adopted. With the energy techniques, we obtain both the second- and fourth-order spectral deferred correction time-marching with local discontinuous Galerkin spatial discretization are unconditional stable. The optimal error estimates for the corresponding fully discrete scheme are derived by the aid of the generalized Gauss–Radau projection. We extend the analysis to problems with higher even-order derivatives. Numerical examples are displayed to verify our theoretical results.

A novel time‐domain numerical methodology for the electromagnetic analysis of an H‐plane tee power divider

A time-domain numerical methodology for the transient analysis of multisection waveguide power dividers is proposed. Waveguide power dividers are essential components of microwave systems for high-power applications. A numerical strategy based on the 3D discontinuous Galerkin time-domain finite element method is used to estimate the electromagnetic propagation inside a power divider with three ports. The proposed approach leads to simulations that are fast, robust, and accurate, this being the result of a parallel local discretisation of the full-wave time-domain Maxwell's equations on unstructured tetrahedral meshes. The stability of the overall numerical scheme is obtained using an upwind numerical flux, an implicit second-order accurate time integration scheme, and Whitney's edge elements. Good agreements are obtained when the computed results are compared with other HFSS simulation results, and with measurement results found in the literature.

A fractional-step DG-FE method for the time-dependent generalized Boussinesq equations

In this work a fractional-step DG-FE method for the time-dependent generalized Boussinesq equations is proposed and analysed. The scheme is composed of two steps. In the first step the original problem is reduced into several scalar elliptic equations. An intermediate velocity and temperature are solved simultaneously. Then in the second step, the incompressibility constraint is enforced and velocity is corrected to be discretely divergence free. Moreover, the introduced elliptic term in the correction step enables the imposition of correct Dirichlet boundary conditions at each temporal step, avoiding the artificial boundary layer introduced by classical pressure correction method. DG-FE discretization strategy is utilized, in which the discontinuous Galerkin spacial discretization for flow equations is employed to obtain local mass conservation and traditional finite element spacial discretization is adopted for heat equation to reduce degrees of freedom. By choosing different symmetry and penalty parameters, SIPG-FE and NIPG-FE methods can be utilized. The consistency and stability of both methods are proved. Preliminary error estimates proving the optimal spacial order and suboptimal temporal order are carried out. Based on a different error equation and the preliminary error estimates, the optimal temporal convergence order is obtained. Numerical tests including a benchmark simulating square cavity flow are then presented, to verify the theoretical analysis and validate the method.

An implicit symplectic solver for high-precision long-term integrations of the Solar System

We present FCIRK16, a 16th-order implicit symplectic integrator for long-term high-precision Solar System simulations. Our integrator takes advantage of the near-Keplerian motion of the planets around the Sun by alternating Keplerian motions with corrections accounting for the planetary interactions. Compared to other symplectic integrators (the Wisdom and Holman map and its higher-order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high-precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of roundoff errors. In addition, unlike typical explicit symplectic integrators for near-Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non-necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point-mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high-order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required.