Explicit Discretization Research Articles

Application of the Entropically Damped Artificial Compressibility model to direct numerical simulation of turbulent channel flow

The feasibility of the Entropically Damped Artificial Compressibility (EDAC) model for simulation of wall-bounded turbulence is examined. A favourable feature of EDAC is the purely parabolic type of governing equations resulting from the introduction of a damping term to the pressure evolution equation. This significantly reduces noise in the velocity divergence field when central schemes are applied for the spatial discretisation. The method of lines is used for the solution of the EDAC equations. A conservative finite difference method of purely 2nd- and mixed 4th/2nd-order is applied. The explicit 4th-order, six-stage low-storage Runge–Kutta method is used for time advancement. As expected, the mixed scheme is superior to the 2nd-order one in terms of both accuracy and computational efficiency.For the first time, the EDAC model is assessed for the direct numerical simulation of wall-bounded turbulent flow requiring a non-uniform mesh. As a particular test case we have chosen the channel flow at the friction Reynolds numbers Reτ=180 and 395. A very good agreement with the reference data is obtained, as documented by the chosen one-point velocity statistics: the mean and r.m.s. profiles and the budgets of the Reynolds stresses. The use of the explicit time discretisation and local (rather than compact) spatial schemes results in easily and efficiently parallelisable solution algorithm. We achieved very high computational performance on a desktop computer when solving the EDAC equations using a code dedicated for the graphics processing unit (GPU). The aspects of the GPU implementation are discussed.

Development of meshless phase field method for two-phase flow

The purpose of the present paper is the development of meshless diffuse approximate method in connection with the phase field formulation for solution of two-phase flow problems. The numerical approach enables single-domain, fixed-node treatment of the involved moving boundaries. The problem is formulated for Newtonian fluids based on a physical model that involves a coupled set of Navier–Stokes and Cahn–Hilliard equations. Diffuse approximate method is structured with a second order polynomial basis, Gaussian weight function, adaptive upwind scheme and eleven-node local subdomain. The pressure-velocity coupling is performed by the incremental fractional step method and explicit time discretization is used to solve the governing equations. The method is demonstrated in axisymmetry for a problem of two co-flowing immiscible fluids with different material properties that yield dripping or jetting of the core fluid. An assessment of the novel method is carried out based on comparison of the present results with the finite volume method outcome. A sensitivity study of various process parameters is carried out and verified against the previously published results. The paper represents a pioneering development in simulation of two-phase flows by a meshless method.

Local Discontinuous Galerkin Method with Implicit–Explicit Time Marching for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we shall present two fully-discrete local discontinuous Galerkin methods, coupled with multi-step implicit–explicit time discretization up to second order, for solving the two-dimensional incompressible miscible displacement problem. To avoid the solving of nonlinear algebraic systems, the extrapolation linearization is adopted to diffusion–dispersion tensor. Under weak temporal-spatial conditions, the optimal error estimates in $$L^{\infty }(L^{2})$$ norm for both concentration and velocity are derived. Numerical experiments are also given to demonstrate the theoretical results.

Optimal Order Error Estimates for Discontinuous Galerkin Methods for the Wave Equation

In this paper, we derive optimal order error estimates for spatially semi-discrete and fully discrete schemes to numerically solve the second-order wave equation. The numerical schemes are constructed with the discontinuous Galerkin (DG) discretization for the spatial variable and the centered second-order finite difference approximation for the temporal variable. Under appropriate regularity assumptions on the solution, the schemes are shown to enjoy the optimal order error bounds in terms of both the spatial mesh-size and the time-step. In Grote and Schotzau (J Sci Comput 40:257–272, 2009), a fully discrete DG scheme is studied with an explicit finite difference temporal discretization where a CFL condition is required on the mesh-size and the time-step, and optimal order error estimates are derived in the $$L^2(\Omega )$$ -norm. In comparison, for our fully discrete DG schemes, we do not require a CFL condition on the mesh-size and the time-step, and our optimal order error estimates are derived for the $$H^1(\Omega )$$ -like norm and the $$L^2(\Omega )$$ norm. Numerical simulation results are reported to illustrate theoretically predicted convergence orders in the $$H^1(\Omega )$$ and $$L^2(\Omega )$$ norms.

Control Oriented Modelling of a Wind Turbine and Farm

The Matlab/Simulink model of the Supergen (Sustainable Power Generation and Supply) Wind 5 MW exemplar wind turbine, which has been employed by a number of researchers at various institutions and Universities over the last decade, is improved, especially in speed, to facilitate wind farm modelling. Note that wind farm modelling usually involves duplicating wind turbine models, hence the speed of each turbine model is critical in wind farm modelling. The objective is achieved through various stages, including prewarping, implicit and explicit discretisation, and conversion to C. Simulation results are presented to demonstrate that improvement in speed is significant and that the resulting wind turbine model can be used for wind farm modelling more efficiently. It is important to highlight that improvement in speed is achieved without compromising the complexity of the turbine model; that is, each turbine included in a wind farm is not simplified or compromised.

A stabilized POD model for turbulent flows over a range of Reynolds numbers: Optimal parameter sampling and constrained projection

We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the N-term expansion. Second, we propose an a posteriori error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the a posteriori error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.

Time Step Validation Method Research for Low-Prandtl Number Fluid Numerical Simulation

The stability condition of Courant number and diffusion number is proved for an SGSD (stability guaranteed second-order difference) scheme by von Neumann method in implicit and explicit discretization of the one-dimensional convection and diffusion terms. Then, a series of numerical simulations of fluid flow and heat transfer based on two-dimensional unsteady state model is used to study the combined natural and MHD (magnetohydrodynamics) convection in a Joule-heated cavity using the finite volume methods, for the fluid of Pr = 0.01, also we use an SGSD scheme and IDEAL (inner doubly iterative efficient algorithm for linked equations) algorithm. It is found that periodic oscillation flow evolves.We propose a new convergence concept for the simulation oscillation results; the results of the numerical experiments are presented and they confirm our theoretical conclusions. The convergence result is checked in another way. It is found that the two approaches have the same results and can judge the validity of the time step. The proposed method is helpful to get reliable results efficiently.

On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme

ABSTRACTIn this work, we consider a damped hyperbolic partial differential equation in multiple spatial dimensions with spatial partial derivatives of non-integer order. The equation under investigation is a fractional extension of the well-known sine-Gordon and Klein–Gordon equations from relativistic quantum mechanics. The system has associated an energy functional which is conserved in the undamped regime, and dissipated in the damped case. In this manuscript, we restrict our study to a bounded spatial domain and propose an explicit finite-difference discretization of the problem using fractional centred differences. Together with the scheme, we propose an approximation for the energy functional and show that the energy of the discrete system is conserved/dissipated when the energy of the continuous model is conserved/dissipated. The method guarantees that the energy functionals are positive, in agreement with the continuous counterparts. We show in this work that the method is a uniquely solvable, consistent, stable and convergent technique.

Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction–diffusion problems

In this paper, an adapted numerical scheme for reaction–diffusion problems generating periodic wavefronts is introduced. Adapted numerical methods for such evolutionary problems are specially tuned to follow prescribed qualitative behaviors of the solutions, making the numerical scheme more accurate and efficient as compared with traditional schemes already known in the literature. Adaptation through the so-called exponential fitting technique leads to methods whose coefficients depend on unknown parameters related to the dynamics and aimed to be numerically computed. Here we propose a strategy for a cheap and accurate estimation of such parameters, which consists essentially in minimizing the leading term of the local truncation error whose expression is provided in a rigorous accuracy analysis. In particular, the presented estimation technique has been applied to a numerical scheme based on combining an adapted finite difference discretization in space with an implicit–explicit time discretization. Numerical experiments confirming the effectiveness of the approach are also provided.

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L^2 ($\Omega$)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

Finit element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for linear convex quadrilaterals of cubic order Serendipity and Lagrange families

This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique, In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be then applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.

Implicit Positivity-Preserving High-Order Discontinuous Galerkin Methods for Conservation Laws

Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been extensively studied in the last several years, but nearly all the developed schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the constraint for the Courant--Friedrichs--Lewy (CFL) number. This makes explicit methods impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this paper we develop a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. We focus on one spatial dimension. However, the result easily generalizes to multidimensional tensor product meshes and polynomial spaces. This work is based on a generalization of the positivity-preserving limiters in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091--312...

A Tetrahedron-based Heat Flux Signature for Cortical Thickness Morphometry Analysis.

Cortical thickness analysis of brain magnetic resonance images is an important technique in neuroimaging research. There are two main computational paradigms, namely voxel-based and surface-based methods. Recently, a tetrahedron-based volumetric morphometry (TBVM) approach involving proper discretization methods was proposed. The multi-scale and physics-based geometric features generated through such methods may yield stronger statistical power. However, several challenges, such as the lack of well-defined thickness statistics and the difficulty in filling tetrahedrons into the thin and curvy cortex structure, impede the broad application of TBVM. In this paper, we present a universal cortical thickness morphometry analysis approach called tetrahedron-based Heat Flux Signature (tHFS) to address these challenges. We define the tetrahedron-based weak form heat equation and Laplace-Beltrami eigen decomposition and give an explicit FEM-based discretization formulation to compute the tHFS. We further show a tHFS metric space with which cortical morphometric distances can be directly visualized. Additionally, we optimize the cortical tetrahedral mesh generation pipeline and fill dense high-quality tetrahedra in the grey matters without sacrificing data integrity. Compared with existing cortical thickness analysis approaches, our experimental results of distinguishing among Alzheimer's disease (AD), cognitively normal (CN) and mild cognitive impairment (MCI) subjects shows that tHFS yields a more accurate representation of cortical thickness morphometry. The tHFS metric experiment provides a more vivid visualization of tHFS's power in separating different clinical groups.

Phase field simulation of Rayleigh–Taylor instability with a meshless method

The purpose of this paper is a numerical study of Rayleigh–Taylor instability problem in two dimensions, based on phase-field (PF) formulation and diffuse approximate method (DAM) meshless solution procedure, enabling single-domain fixed-node approach for coping with moving boundary problems. The problem is formulated based on three physically different models that reduce to solving Cahn–Hilliard equation in addition to the Navier–Stokes equations for incompressible fluids. The governing equations are solved by using explicit time discretization. DAM is structured with second order polynomial basis, Gaussian weighting, upwinding and local domain support. The pressure–velocity coupling is performed by the fractional step method. The assessment of the method is carried out based on different node density, weighting, and the size of the local domain support. The novel approach is verified by reproducing the boundary dynamics, consistent with the previously published results. A detailed comparison with the volume of fluid finite volume approach is presented. The combination of PF and DAM provides a valuable numerical tool for solving immiscible convective hydrodynamics problems. The paper represents a pioneering attempt in solution of Rayleigh–Taylor instability problem by a meshless solution of the phase-field formulation of the problem.

An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions

In this work, we investigate numerically a model governed by a multidimensional nonlinear wave equation with damping and fractional diffusion. The governing partial differential equation considers the presence of Riesz space-fractional derivatives of orders in (1, 2], and homogeneous Dirichlet boundary data are imposed on a closed and bounded spatial domain. The model under investigation possesses an energy function which is preserved in the undamped regime. In the damped case, we establish the property of energy dissipation of the model using arguments from functional analysis. Motivated by these results, we propose an explicit finite-difference discretization of our fractional model based on the use of fractional centered differences. Associated to our discrete model, we also propose discretizations of the energy quantities. We establish that the discrete energy is conserved in the undamped regime, and that it dissipates in the damped scenario. Among the most important numerical features of our scheme, we show that the method has a consistency of second order, that it is stable and that it has a quadratic order of convergence. Some one- and two-dimensional simulations are shown in this work to illustrate the fact that the technique is capable of preserving the discrete energy in the undamped regime. For the sake of convenience, we provide a Matlab implementation of our method for the one-dimensional scenario.

An implicit algorithm of solving Navier–Stokes equations to simulate flows in anisotropic porous media

A coupled computational algorithm for modified Navier–Stokes equations to simulate flows in anisotropic porous media is proposed and described. The difference from the classic SIMPLE algorithm is in the completely implicit relationship between velocity and pressure owing to the implicit terms of the pressure and mass flow gradients in the continuity equation and momentum equation. One of the attractive features of this algorithm is the possibility of completely implicit discretization of off-diagonal components of the porous medium resistance tensor in the right-hand side of the momentum equation. Implicit discretization allows reducing the number of linear iterations as compared to the SIMPLE algorithm with explicit discretization of off-diagonal components. The specific features of discretization of modified equations including the discretization of boundary conditions and components of the porous medium resistance tensor are considered. The proposed algorithm is verified in comparison with the SIMPLE algorithm on a series of benchmark problems, such as the problem of a flow through a porous insert, a flow in a divided channel, and a flow through a cylindrical porous filter. The total problem runtimes and the number of iterations required for complete convergence are given to compare the two algorithms.

Flux-corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements

The Flux-Corrected Transport (FCT) algorithm is applied to the unsteady and steady-state particle transport equation. The proposed FCT method employs the following: (1) a low-order, positivity-preserving scheme, based on the application of M-matrix properties, (2) a high-order scheme based on the entropy viscosity method introduced by Guermond [1], and (3) local, discrete solution bounds derived from the integral transport equation. The resulting scheme is second-order accurate in space, enforces an entropy inequality, mitigates the formation of spurious oscillations, and guarantees the absence of negativities. Space discretization is achieved using continuous finite elements. Time discretizations for unsteady problems include theta schemes such as explicit and implicit Euler, and strong-stability preserving Runge–Kutta (SSPRK) methods. The developed FCT scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady-state and implicit time discretizations.

Numerical simulation of Bloch equations for dynamic magnetic resonance imaging

Magnetic Resonance Imaging (MRI) is a widely applied non-invasive imaging modality based on non-ionizing radiation which gives excellent images and soft tissue contrast of living tissues. We consider the modified Bloch problem as a model of MRI for flowing spins in an incompressible flow field. After establishing the well-posedness of the corresponding evolution problem, we analyze its spatial semi-discretization using discontinuous Galerkin methods. The high frequency time evolution requires a proper explicit and adaptive temporal discretization. The applicability of the approach is shown for basic examples.

Concurrent development of local and non-local damage with the Thick Level Set approach: Implementation aspects and application to quasi-brittle failure

The present paper focuses on the discretization and implementation of the Thick Level Set (TLS) approach for quasi-brittle failure when dealing with both local and non-local development of damage.The TLS damage model introduces a length scale and nonlocality in the Fracture Process Zone (FPZ) and it handles the continuous to discontinuous failure transition. The added nonlocality is limited to the FPZ and therefore the model is local away from the FPZ or before its emergence. It leads to concurrent development of local and nonlocal damage during simulations. This paper presents a space–time discretization that handles concurrent computations in a unified manner. It is based on an explicit time discretization written as a predictor–corrector scheme and a space discretization that uses specific approximation functions (modes) that embed nonlocality. It also uses an alternative implementation of the eXtended Finite Element Method (X-FEM) enrichment used in the TLS which is based on virtual nodes.

Enabling Highly Efficient Spectral Discretization-Based Eigen-Analysis Methods by Kronecker Product

This letter presents a computationally efficient approach to implement the spectral discretization-based methods for eigenanalysis of large delayed cyber-physical power system by intensively exploiting the unique property of Kronecker product. Theoretical analyses and numerical tests demonstrate the novelty and effectiveness of the efficiently implemented explicit infinitesimal generator discretization method.