Accurate Discretization Research Articles

Numerical simulation of rarefied supersonic flows using a fourth-order maximum-entropy moment method with interpolative closure

Maximum-entropy moment methods allow for the modeling of gases from the continuum regime to strongly rarefied conditions. The development of approximated solutions to the entropy maximization problem has made these methods computationally affordable. In this work, we apply a fourth-order maximum-entropy moment method to the study of supersonic rarefied flows. For such conditions, we compare the maximum-entropy solutions to results obtained from the kinetic theory of gases at different Knudsen numbers. The analysis is performed for both a simplified model of a gas with a single translational degree of freedom (5-moment system) and for a typical gas with three degrees of freedom (14-moment system). The maximum-entropy method is applied to the study of the Sod shock-tube problem at various rarefaction levels, and to the simulation of two-dimensional low-collisional crossed supersonic jets. We show that, in rarefied supersonic conditions, it is important to employ accurate estimates of the wave speeds. Since analytical expressions are not presently available, we propose an approximation, valid for the 14-moment system. In these conditions, the solution of the maximum-entropy system is shown to realize large degrees of non-equilibrium and to approach the Junk subspace, yet provides a good overall accuracy and agreement with the kinetic theory. Numerical procedures for reaching second-order accurate discretizations are discussed, as well as the implementation of the 14-moment solver on Graphics Processing Units (GPUs).

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

Low-Thrust Transfer Orbit Optimization Using Sequential Convex Programming and Adaptive Mesh Refinement

In this paper, a mesh-adaptive Hermite–Simpson sequential convex programming (SCP) method is proposed for solving low-thrust orbit transfer problems efficiently and accurately. First, we develop a Hermite–Simpson convex programming method that utilizes quadratic polynomial control interpolation to improve discretization accuracy. Then, a mesh-adaptive Hermite–Simpson SCP method is proposed by incorporating an improved adaptive mesh refinement method into the Hermite–Simpson SCP method for solving optimal control accurately and efficiently. The proposed method is applied to two-dimensional time-optimal and three-dimensional fuel-optimal low-thrust orbit transfer problems to show its features, and its performance is compared against several existing methods. Numerical simulations show that the proposed method achieves more accurate solutions while being computationally more efficient compared to the trapezoidal SCP method. In comparison to the GPOPS-II software, the proposed method produces solutions of comparable accuracy but is more computationally efficient.

VOF method in two-stage fourth order time-stepping framework

The fraction volume of fluid (VOF) method is a successful approach to track the dynamics of free boundaries thanks to its flexibility, efficiency and simplicity. And various efficient numerical methods have been developed to solve the associated VOF equation. This paper applies the two-stage fourth order accurate time discretization in [21] for the evolution of the VOF equation, aiming to provide a compact diffuse-interface algorithm simple for implementation, in order to incorporate with CFD shock-capturing schemes for the simulation of multi-material/phase problems with sharp interfaces. As such, the algorithm is decoupled from those for fluid flows as long as the velocity of the interface is given. Numerical results demonstrate the high resolution and high-fidelity for the interface capturing.

A new open-source library based on novel high-resolution structure-preserving convection schemes

This work develops a new finite volume convection scheme and open-source library for convection-dominated problems on unstructured grids. The formulated reconstruction operators can realize high-resolution and structure-preserving properties by adjusting numerical dissipation and anti-dissipation errors in a unified normalized-variable diagram. Then, this work develops a new open-source library based on the devised schemes within one of the most widely used general-purpose open-source software whose solution quality is intrinsically limited by the low accuracy of the conventional total variation diminishing schemes. The performance of the new library is first verified by some fundamental problems including pure advection and inviscid Euler equations. The numerical results of the accuracy tests show that the developed interpolation library can significantly reduce numerical errors with a minor increased computational cost compared with the conventional second-order schemes. The complex profile advection test demonstrates that the new library can offer an improved structure-preserving property that gives essentially oscillation-free solutions and preserves the structures of the passive transported scalar. Furthermore, the robustness of the new library is validated by compressible high-speed flow simulations with irregular geometries. Thus, this work presents a new library of high-resolution structure-preserving schemes for the accurate and efficient discretization of the convection term which plays an essential role in a wide range of computational fluid dynamics. The source code of the new scheme and library are released via DOI:10.5281/zenodo.8399000.

Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time

In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations.

A First collision source method using spherical harmonics method for arbitrary shape and order finite element mesh

The discrete ordinates (SN) method is a highly effective and accurate angular discretization method for the neutron transport equation, but suffers from the inherent ray effect in low-scattering regions. The first collision source (FCS) method employing the methods without ray effect to obtain the uncollided flux is a common ray effect mitigation technique. In this paper, a novel FCS method using the spherical harmonics (PN) method to calculate the uncollided flux is proposed. The PN for the first collision source (PN-FCS) method possesses distinctive advantages in several aspects: (1) it can be applied in the problem containing reflective boundary conditions that are difficult to handle with traditional FCS method, (2) it uses the discontinuous Galerkin method to perform spatial discretization, and thus can be easily applied in arbitrary shape and order finite element meshes, (3) it can be effectively calculated in parallel on distributed meshes, (4) the accuracy and computational cost can be flexibly adjusted. Besides, the filter technique is adopted to improve the performance of the PN method in stream-dominated regions. The PN-FCS method has been implemented in two- and three-dimensional neutron transport code and tested by various testing problems. Numerical results show that the PN-FCS method can effectively mitigate the ray effect.

Solid-liquid rotary kilns: An experimental and CFD-DEM study

Experiments for the validation of unresolved CFD-DEM software for solid–liquid flows are often expensive, time consuming and generally provide little insight into the local particles dynamics. Additionally, several DEM parameters such as the particle surface properties are often obtained from experiments in air and used for all CFD-DEM simulations even when the fluid is a liquid. We design and perform a simple and time efficient gas–solid and liquid–solid rotary kiln benchmark for the purpose of creating a validation case for unresolved CFD-DEM software which we use to validate the unresolved CFD-DEM model of the open-source software Lethe. This case, which contains dense solid–solid contacts and strong solid–fluid forces gives insight on the importance of proper calibration of DEM surface properties in solid–liquid flows as well as on the importance of the lift force. Furthermore, it is highly sensitive to the accuracy of the CFD discretization.

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems

AbstractFully implicit Runge–Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this work, we consider strongly A‐stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures. For the arising large algebraic systems we introduce efficient preconditioners, that (1) use only real arithmetic, (2) demonstrate robustness with respect to problem and discretization parameters, and (3) allow for fully stage‐parallel solution. The preconditioners are based on the observation that the lower‐triangular part of the coefficient matrices in the Butcher tableau has larger in magnitude values, compared to the corresponding strictly upper‐triangular part. We analyze the spectrum of the corresponding preconditioned systems and illustrate their performance with numerical experiments. Even though the observation has been made some time ago, its impact on constructing stage‐parallel preconditioners has not yet been done and its systematic study constitutes the novelty of this article.

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

AbstractWe propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation

We consider the numerical approximations of the viscous Cahn–Hilliard equation with either the Ginzburg–Landau polynomial potential or Flory–Huggins logarithmic potential. One challenge in solving such a fourth-order-in-space system is to develop accurate temporal discretization to preserve the energy stability, mass conservation, and maximum principle. We resolve this issue by developing a family of first- and second-order time marching schemes based on exponential time differencing Runge–Kutta (ETDRK) methods. We prove that the proposed schemes unconditionally preserve the three characteristics of the viscous Cahn–Hilliard equation. Because of the preservation of maximum principle, an error estimate in the infinity-norm is derived for the fully discrete system. Various numerical experiments are performed to verify the theoretical results and demonstrate the excellent performance of the proposed schemes.

Numerical Treatment of State‐Dependent Permeability in Multiphysics Problems

AbstractConstitutive laws relating fluid potentials and fluxes in a nonlinear manner are common in several porous media applications, including biological and reactive flows, poromechanics, and fracture deformation. Compared to the standard, linear Darcy's law, such enhanced flux relations increase both the degree of nonlinearity, and, in the case of multiphysics simulations, coupling strength between processes. While incorporating the nonlinearities into simulation models is thus paramount for computational efficiency, correct linearization, as is needed for incorporation in Newton's method, is challenging from a practical perspective. The standard approach is therefore to ignore nonlinearities in the permeability during linearization. For finite volume methods, which are popular in porous media applications, complete linearization is feasible only for the simplest flux discretization, namely the two‐point flux approximation. We introduce an approximated linearization scheme for finite volume methods that is exact for the two‐point scheme and can be applied to more advanced and accurate discretizations, exemplified herein by a multi‐point flux stencil. We test the new method for both nonlinear porous media flow and several multiphysics simulations. Our results show that the new linearization consistently outperforms the standard approach. Moreover our scheme achieves asymptotic second order convergence of the Newton iterations, in contrast to the linear convergence obtained with the standard approach.

Trajectory Optimization of Long-Range Guided Rocket-Propelled Vehicle Based on Pseudo-spectral Sequential Convex Programming

Aiming at the trajectory optimization problem of long-range guided rocket-propelled vehicles with multiple constraints, an improved pseudo-spectral sequence convex programming, which takes into account the efficiency and accuracy of optimization, is proposed. First, taking the change rate of the angle of attack and the bank angle as the control variables, the vehicle’s dynamic equation is discretized by using the flipped Radau pseudo-spectral method, and the trajectory optimization problem including start, end, and path constraints is transformed into a convex optimization problem by using the continuous linearization method, and then the solution is obtained by using the original dual interior point method. To make the optimized trajectory near the no-fly zone or the severe weather area more accurate, the concept of the predicted separation point is introduced to subdivide the trajectory. The simulation results show that the improved pseudo-spectral sequence convex programming has high discretization accuracy and fast solution speed, and can be applied to the online use of long-range guided rocket-propelled vehicles.

On a Fast Hough/Radon Transform as a Compact Summation Scheme over Digital Straight Line Segments

The Hough transform, interpreted as the discretization of the Radon transform, is a widely used tool in image processing and machine vision. The primary way to speed it up is to employ the Brady–Yong algorithm. However, the accuracy of the straight line discretization utilized in this algorithm is limited. In this study, we propose a novel algorithm called ASD2 that offers fast computation of the Hough transform for images of arbitrary sizes. Our approach adopts a computation scheme similar to the Brady–Yong algorithm but incorporates the best possible line discretization for improved accuracy. By employing the Method of Four Russians, we demonstrate that for an image of size n×n where n=8q and q∈N, the computational complexity of the ASD2 algorithm is O(n8/3) when summing over O(n2) digital straight line segments.

Discrete-Time Current Control of Salient Machines With a Simplified Model

The discrete-time model for salient machines with accurate discretization is quite complicated due to different <i>dq</i>-axes inductances. This paper proposes an approximation for the discrete-time model with high accuracy and efficient calculation. Various patterns of pulse-width-modulation updating and current sampling that result in one-sample or a-half-sample computational delay are studied, and corresponding current controllers are directly designed in the discrete-time domain with the simplified model. Decoupled responses between <i>dq</i>-axes are realized in reference tracking. Active resistance is also employed to improve the disturbance rejection. The experimental verification is performed on an interior permanent magnet synchronous machine.

Isogeometric degenerated shell formulation for post-buckling analysis of composite variable-stiffness shells

Variable-stiffness (VS) laminated shells with curved fibers have gradually become a promising lightweight structural concept in the aerospace field. However, as the complexity of the shell geometry and the fiber angle increases, the traditional finite element method (FEM) requires mesh refinement to guarantee accurate discretization, which significantly increases the computational cost, especially in the post-buckling analysis with multiple solving requirements. In this study, a non-uniform rational B-Splines (NURBS)-based degenerated solid shell formulation in the total Lagrangian framework is proposed for the post-buckling analysis of VS laminated shells. Since the nonlinear function related to the rotational degrees of freedom (DOFs) is additionally introduced into the displacement field expression, the restriction on the magnitude of the nodal rotations is effectively eliminated. In addition, the paper adopts the NURBS surface reconstruction technology based on the point-line-surface features to build B-spline CAD models from the point cloud with geometric imperfect features. Then, an integrated modeling-analysis framework for predicting the post-buckling behavior of VS structures with geometric imperfections is proposed. A series of numerical examples systematically demonstrate that the framework can effectively conduct the post-buckling analysis of shell structures with geometric imperfections. Meanwhile, the accuracy and robustness of the solutions by IGA are proved to be better than FEM.

Immersed boundary method for the complete electrode model in electrical impedance tomography

We propose an immersed boundary scheme for the numerical resolution of the Complete Electrode Model in Electrical Impedance Tomography, that we use as a main ingredient in the resolution of inverse problems in medical imaging. Such method allows to use a Cartesian mesh without accurate discretization of the boundary, which is useful in situations where the boundary is complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency with two dimensional direct and inverse problems.

Modeling and inversion in acoustic-elastic coupled media using energy-stable summation-by-parts operators

Seismic data acquired at the seafloor are valuable in characterizing the subsurface and monitoring producing hydrocarbon fields. To fully use such data, a stable, accurate, and efficient numerical scheme is needed that accounts for acoustic and elastic wave propagation, their interaction at the seafloor interface, and for sources and receivers placed on either side of that interface. Existing methods either make incorrect assumptions or have high implementation and computational costs. We have developed a high-order finite-difference summation-by-parts framework for the acoustic-elastic wave equations in second-order form (in terms of displacements in the solid and an extension of velocity potential in the fluid). Our modified discretization of the elastic operator overcomes the dispersion errors known to plague displacement-based schemes in the high VP/ VS limit. We weakly impose boundary and interface conditions using simultaneous approximation terms, leading to an energy-stable numerical scheme that rigorously handles point injection and extraction. The fully discrete system is self-adjoint after a time reversal and a sign flip and is furthermore a high-order accurate discretization of the continuous problem. The self-adjointness ensures that forward and adjoint wavefields are computed with similar accuracy and simplifies gradient computation for inversion purposes. We find that our numerical scheme achieves accuracy comparable to a more computationally expensive spectral-element method and demonstrate its application to full-waveform inversion using the Marmousi2 model.

Assessment of solving the RANS equations with two-equation eddy-viscosity models using high-order accurate discretization

A challenge that faces high-order methods for industrial applications generally is turbulence modeling at high Reynolds numbers. Large eddy simulation is studied extensively for high-order methods, nevertheless, its computational cost is enormous for industrial applications. The hybrid LES/RANS compromises the computational cost and the modeling error, however, solving the Reynolds-averaged Navier-Stokes equations is a resilient task for high-order methods, due to the non-smooth profiles of the turbulence quantities. Taking into account the complexity with high-order methods and the fairly large modeling errors of the RANS modeling, low-order methods has proved to be more pragmatic. For instance, in the discontinuous Galerkin framework, the polynomial approximation for these quantities leads to large oscillations that obstructs the non-linear solver. To use the high-order methods for industrial cases it is essential to have reliable implementations of two-equation turbulence models in RANS formulations. In this paper, a RANS discretization based on hybridizable discontinuous Galerkin is presented for the standard, TNT, BSL and SST versions of the k−ω model for applications at Reynolds numbers up to 109. A particular focus is given to the treatment of the specific rate of turbulence dissipation ω in the high-order framework. The complexity increases with these types of models as the value of ω goes to infinity at solid walls. Additionally, with minor modifications to the numerical flux definition, the turbulence model formulation can be solved by discontinuous Galerkin method as well. The results show remarkable improvements regarding the error magnitudes and non-linear convergence rate (iterative error) compared to second-order finite volume based solvers.