Spatial Discretization Schemes Research Articles

Element‐based coupling modeling of peridynamics and classical continuum mechanics for dynamic brittle fracture

AbstractIn this work, the hybrid peridynamic and classical continuum mechanical model based on the element‐type numerical discretization is used to study the dynamic fracture problem of brittle materials. The weak form of the motion equation for the hybrid model is derived by applying Hamilton's principle. An element‐based spatial discretization scheme is set up using the peridynamics‐based finite element method (PeriFEM) (Han and Li). The ‐convergence and m‐convergence of the hybrid model are analyzed to verify the numerical convergence and the mesh dependency. The influence of the area of the peridynamic model subdomain and the hybrid model subdomain on the calculation efficiency and accuracy is discussed. And the relationship between the crack propagation speed and the crack branching phenomenon is also investigated. Good agreements are obtained by comparing the crack path and the crack speed profile predicted by the hybrid model with those in the literature. Numerical results verify the ability and efficiency of the hybrid model for dynamic fracture problem.

Validation of a Human Upper Airway Computational Fluid Dynamics Model for Turbulent Mixing

Abstract Validation of a computational fluid dynamics (CFD) model used to simulate turbulent exchange in an anatomically detailed human upper airway with realistic breathing states is provided. Proper model validation is vital in confirming that temporal mixing and species distribution are accurate, therefore making the model useful in generalized turbulent mixing studies of the upper airway. Numerous levels of refinement were tested for time-step and mesh independence. Higher and lower rigor groups of modeling methodologies involved spatial discretization schemes, gradient reconstruction methods, transient formulations, and turbulence frameworks. A dual mesh independence study revealed that the rate of approach to mesh independence is a function of computational rigor and that multiple mesh independence studies should be carried out in parallel. The final validated model consisted of the finest mesh used in this study (8 × 106 cells), a time-step equating to 4000 timesteps per breath cycle, and higher rigor modeling methodologies. While its results were within the acceptable deviation from the experimental data, it was not as close as the model that utilized the coarsest mesh (∼2 × 106 cells), the fewest timesteps per breath cycle (128 timesteps per breath cycle), and lower rigor methodologies. Though the latter model was closer to the experimental data, it was proven to not be numerically independent, highlighting the importance of utilizing a myriad of metrics to prove numerical independence. Restricting independence studies to only using metrics from experimental comparisons is insufficient for proper validation.

Computational Study of Supersonic Flow Over a Flat Plate With Protrusion

Motivated by a developmental approach of using micro-actuated surface protrusions to control/maneuver slender bodies in supersonic flight, this paper presents a detailed study of a two dimensional laminar supersonic flow over a flat plate with a surface protrusion. The flow field is computed by solving the Navier-Stokes equations using the finite difference method with the particle velocity upwinding scheme (PVUS) for spatial discretization, and the explicit Mac Cormack scheme for temporal integration. A range of free stream Mach numbers (2.0 - 4.5), Reynolds numbers (1000 - 100000) and protrusion heights (0.866% to 8.66% of the characteristic length) has been considered for a thorough parametric study. The parametric study indicates that the oblique shock structure and strength are influenced by all the variables to varying extents. However, the shock location is substantially altered by the protrusion height, Reynolds number, and protrusion shape, while the influence of Mach number is only marginal. An increase in the protrusion height results in an increased wall pressure as well as increased separation lengths on both sides of the protrusion. In contrast, an increase in Mach number increases the wall pressure, but moves the separation point marginally towards the protrusion. Increase in Reynolds number and protrusion bluffness (triangular → trapezoidal → rectangular) increases separation lengths on both sides of the protrusion. In addition to taking a close look at the flow physics, particularly in the protrusion vicinity, considerable attention has also been devoted to important design parameters such as wall pressure, skin friction and the overall normal and tangential force coefficients.

CFD Simulation of Low Speed Turbulent Flow Problems Using Unsteady Rans and Large Eddy Simulation Approach

The present paper focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Navier Stokes equation in an inertial frame of reference for prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressurevelocity solution strategy. Effect of turbulence is simulated using one of the two different approaches. In the Unsteady Reynolds Averaged (URANS) approach coupled to appropriate eddy viscosity based turbulence models, the Navier Stokes (NS) equations, averaged over the whole range of turbulent length scales of the flow, are solved numerically. On the other hand, in the Large Eddy Simulation (LES) approach, the model filtered 3D NS equations are directly solved for the flow variables to resolve the large scale turbulent motions whereas the transport processes at the fine subgrid scale level only are simulated using simple algebraic turbulence model. The capabilities and limitations of both the cost-effective URANS approach and the relatively expensive but rich in physics LES approach have been demonstrated for few application problems of engineering interest.

Global existence of classical solutions and numerical simulations of a cancer invasion model

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. Our theory and numerical developments are accompanied by some simulations in two and three spatial dimensions.

Robust Wireless Rechargeable Sensor Networks

Wireless rechargeable sensor networks have become a hot research issue as it can overcome the limited energy bottleneck of wireless sensor networks owing to the recent breakthrough of wireless power transfer technology. Though network lifetime is prolonged and sensor nodes can sustain immortally, the issue of network robustness is overlooked, yielding most theoretical work unsuitable for practical applications when confronting with unpredictable packet loss. In this paper, we address the network robustness issue by maximizing the charging utility in a risk-averse view. First, we build a risk-averse model based on the concept of CVaR (Conditional Value at Risk), which trades-off charging utility and risk aversion for quantifying robustness. Then, we propose a spatial discretization scheme to construct a charging route for mobile charger, which can reduce computational overhead. Afterwards, a path optimization scheme is designed to further improve the charging utility. We convert the original problem into the submodular function maximization problem and propose a method with a performance guarantee while maximizing the system robustness. Finally, testbed experiments and simulations are conducted, and the results demonstrate that our schemes outperform comparison algorithms by at least 22.4% in effective energy in the presence of risks to guarantee system robustness.

Effect of cube spacings on the three-dimensional flow structure over an array of wall-mounted cube

The turbulent flow over an array of cubes mounted on one of the walls of a channel has been investigated using direct numerical simulation for cube spacing that ranges between 2.0 and 4.0. The Reynolds number based on the cube size and the average streamwise velocity is chosen to be 4000. The Navier–Stokes equations have been discretized using second-order spatial and temporal discretization schemes. The present investigation focuses on the flow structures and comprehensive characterization of the separated zones surrounding the cubes, as well as the associated wall-shear stress. A vortex shedding has been observed for the cube spacings of 3.0 and 4.0 without any evidence of vortex shedding for the lowest pitch of 2.0. For the two cases having pitches of 3.0 and 4.0, the presence of the unsteady separation bubbles at the cube's top and side surfaces results in a decrease in wall-shear stress. The quadrant analysis for the region close to the top surface of the cube is performed with the help of the joint probability density function, which reveals dual peaks within the recirculation bubble at the top surface of the cube for higher cube spacings. By conducting an invariant analysis of the Reynolds stress tensor for different cube spacings, we have explored the characteristic of Reynolds stress anisotropy due to the total fluctuations. The production of negative turbulent kinetic energy (TKE) is observed in different regions within the flow domain, among which the horseshoe vortex region for each cube spacing reveals its dominant presence. The physical mechanism responsible for the production of the negative TKE has also been attempted by decomposing the production term into two parts, namely, normal and shear components.

DNN-HDG: A deep learning hybridized discontinuous Galerkin method for solving some elliptic problems

In this paper, we construct two deep neural network (DNN) approaches based on the hybridized discontinuous Galerkin (HDG) method for solving some elliptic problems. The main idea is to use an HDG scheme for spatial discretization and then estimate solutions using the deep learning idea. In fact, by employing DNN, we can achieve robust methods compared to classical methods for solving noisy and high-dimensional problems. A brief analysis shows that the loss functions corresponding to the proposed methods, which are called DNN-HDG-I and DNN-HDG-II, converge to zero as the mesh step size reduces. By testing several examples in one, two, and three dimensions, we demonstrate the performance of the proposed methods that show their apparent advantages especially compared to classical HDG methods. As we expected, the DNN-HDG methods can efficiently and accurately extract the pattern of the solutions in all three dimensions. To show the superiority of the proposed schemes, the DNN-HDG methods are compared with the classical HDG method for solving a problem involving some noisy data. Specifically, we employ the proposed methods for solving a problem which its exact solution is not accessible and compare it with the HDG solution. In this paper, the uniformly random data and Jacobi polynomial roots are considered, respectively, as the training data on the boundary and quadrature points for approximating integrals. Also, we find that the proposed methods are not required a high number of neurons (at most 30) and hidden layers (at most 10) for satisfactory results with acceptable accuracy.

Fully conservative and pressure-equilibrium preserving scheme for compressible multi-component flows

In compressible flows with a variable ratio of specific heats, such as multi-component flows, the conventional conservative schemes generate spurious pressure oscillations at multi-component interfaces even when the interfaces are smooth. In this study, we propose a novel spatial discretization scheme that maintains both the primary conservation and pressure equilibrium in discontinuity-free compressible multi-component flows. The key to this study is the compatibility condition, which is the discrete condition for implicitly satisfying the pressure equilibrium at the discrete level. The compatibility condition is derived based on the analytical relation between the mass of species and the ratio of specific heats in the governing equations. The proposed scheme is constructed by the spatial discretization that satisfies the compatibility condition and can be applied to the arbitrary number of species components. The inviscid smooth interfaces advection problems verify that the proposed scheme satisfies both the pressure equilibrium and primary conservation only by solving the conservation equations, unlike the existing schemes that solve non-conservative or overspecified equations.

Nonpolynomial Spline Method for Singularly Perturbed Time‐Dependent Parabolic Problem with Two Small Parameters

This study deals with the numerical solution of parabolic convection‐diffusion problems involving two small positive parameters and arising in modeling of hydrodynamics. To approximate the solution, the backward Euler method for time stepping and fitted trigonometric‐spline scheme for spatial discretization are considered on uniform meshes. The resulting scheme is shown to be uniformly convergent and its rate of convergence is one in the time variable and two in the space variable. The accuracy and rate of convergence are enhanced by using the Richardson extrapolation. To support the theoretically shown convergence analysis, we have taken some numerical examples and compared the absolute maximum error of the current method with some methods existing in the literature.

A General Framework to Simulate Diffusions with Discontinuous Coefficients and Local Times

In this article, we propose an efficient general simulation method for diffusions that are solutions to stochastic differential equations with discontinuous coefficients and local time terms. The proposed method is based on sampling from the corresponding continuous-time Markov chain approximation. In contrast to existing time discretization schemes, the Markov chain approximation method corresponds to a spatial discretization scheme and is demonstrated to be particularly suited for simulating diffusion processes with discontinuities in their state space. We establish the theoretical convergence order and also demonstrate the accuracy and robustness of the method in numerical examples by comparing it to the known benchmarks in terms of root mean squared error, runtime, and the parameter sensitivity.

A primal-dual approach for solving conservation laws with implicit in time approximations

In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential equation (PDE) by casting it as a saddle point of a min-max problem and using iterative optimization methods to find the saddle point. Our approach is flexible with the choice of both time and spatial discretization schemes. It benefits from the implicit structure and gains large regions of stability, and overcomes the restriction on the mesh size in time by explicit schemes from Courant–Friedrichs–Lewy (CFL) conditions (really via von Neumann stability analysis). Nevertheless, it is highly parallelizable in principle, and easy-to-implement. In particular, no nonlinear inversions are required! Specifically, we illustrate our approach using the finite difference scheme and discontinuous Galerkin method for the spatial scheme; backward Euler and backward differentiation formulas for implicit discretization in time. Numerical experiments illustrate the effectiveness and robustness of the approach. In future work, we will demonstrate that our idea of replacing an initial-value evolution equation with this primal-dual hybrid gradient approach has great advantages in many other situations.

Alternating direction implicit method for singularly perturbed 2D parabolic convection–diffusion–reaction problem with two small parameters

ABSTRACT In this article, we construct and analyse an Alternating Direction Implicit (ADI) scheme for singularly perturbed 2D parabolic convection–diffusion–reaction problems with two small parameters. We consider the operator-splitting ADI finite difference scheme for time stepping on a uniform mesh and a simple upwind-difference scheme for spatial discretization on a specially designed piecewise-uniform Shishkin mesh. The resulting scheme is proved to be uniformly convergent of order , where N, M are the spatial and temporal parameters respectively. Numerical experiments confirm the theoretical results and the effectiveness of the proposed method.

The Finite Element Numerical Study of Transonic Flow of Moist Air with Nonequilibrium Condensation

The finite element numerical simulation of the transonic flow with condensation is studied in this paper. The transformed weak form is numerically discretized by using the classical Galerkin finite element spatial discretization scheme and the backward Euler difference time discretization scheme. Because the coupled Euler equations are convection equations, the results obtained by the classical Galerkin method often have nonphysical numerical oscillations, so the stabilization method is introduced. Another important feature of compressible Euler equation is the existence of shock wave. Because the thickness of shock wave is small and difficult to capture, the shock wave can be better captured by adding artificial viscosity term. Here, the streamline upwind/Petrov–Galerkin (SUPG) finite element method with good stability and the isotropic diffusion method with shock capture ability are used to solve the problem of transonic flow with condensation. In the simulation of steady problems, in order to improve the calculation accuracy of the existing area of the shock wave, the adaptive mesh refinement technology is added, which refines the mesh in the region of the shock wave to improve the resolution of the shock wave. Numerical experiments verify the feasibility and stabilization of the numerical method. Finally, the simulation of transonic flow around NACA-0012 and parallel-jet nozzle A1 has obtained the better numerical results.

A multi-resolution particle method with high order accuracy for solid-liquid phase change represented by sharp moving interface

This paper develops a highly accurate, multi-resolution particle method to simulate solid-liquid phase change coupled with the thermal flow. Instead of including the latent heat in the governing equation, the heat equations for solid and liquid phases are solved separately. A sharp interface model is proposed to represent the solid-liquid interface explicitly. The sharp interface, represented by discrete nodes, provides the Neumann boundary condition for pressure and the Dirichlet boundary condition for velocity/temperature, respectively. Based on temperature gradients in the solid and liquid phases, the positions of these interface nodes are updated every time step. The Eulerian-based formulation, rather than the conventional Lagrangian-based one, is utilized to minimize time step-dependent error. Up to 4th order spatial discretization scheme is adopted based on the Least Square Moving Particle Semi-implicit (LSMPS) scheme. Moreover, a geometry-based multi-resolution scheme is introduced to dynamically refine the spatial resolution near the interface for saving computational cost. The 1-D Stefan problem is firstly simulated to verify the accuracy of the proposed sharp interface model. Then, the consistency of the multi-resolution scheme is investigated by a convergence study of the Taylor-Green vortex problem. After that, numerical simulations of natural convection in a cavity are performed with different spatial resolutions and high order schemes. Resulted computational costs are compared and discussed. Finally, the problems of melting by natural convection with different Rayleigh numbers are investigated. The results achieved so far indicate that the multi-resolution and high order schemes have great potential to save computational cost.

Research on flow field characteristics of nozzle plume for a solid rocket motor

In order to explore the flow field characteristics of nozzle plume for a small solid rocket motor (SRM), the two-dimensional axisymmetric N-S governing equations with component transports and chemical reaction terms are established. Then, the finite volume method of AUSM (Advection Upstream Splitting Method) spatial discretization scheme is applied for numerical discretization, and its solution is solved by time iteration to make the plume flow field steady. This technique is applied to research the flow field characteristics and characteristic parameter distributions of nozzle plume under different inflow conditions. The results are shown that: when considering the reburning reactions of nozzle plume, the temperature of the gas jet core area is much higher than the flow field temperature that the reburning phenomenon is not considered. In addition, the reburning phenomenon mainly occurs in the air-gas mixing and gas development areas, but in the initial core area of gas jet, this phenomenon is not obvious. What’s more, with the increase of altitude, the influence area of the gas jet gradually becomes larger. However, as the incoming flow Mach number increases, the intensity of the change of gas temperature along the nozzle axis weakens, and also the number of wave nodes and the influence region of gas jet gradually decrease.

Comprehensive analysis of entropy conservation property of non-dissipative schemes for compressible flows: KEEP scheme redefined

A theoretical analysis of the entropy conservation properties is conducted to explain the different behaviors of the non-dissipative finite-difference spatial discretization schemes, such as the kinetic-energy and entropy preserving (KEEP) schemes. The analysis is conducted based on the spatially-discretized entropy-evolution equation derived from the Euler equations with retaining the discrete-level strictness. The present analysis shows that the analytical relations (ARs) employed by the KEEP schemes eliminate some terms of the discretized entropy-evolution equation and help simplify the equation, while the ARs are not sufficient to explain the entropy-conservation property. Therefore, the entropy error is decomposed into several terms, and the behaviors of those decomposed error terms are observed in the compressible inviscid Taylor–Green vortex test case. The results of the test case show that the terms containing the velocity difference between two grid points cause significant entropy conservation error and result in the different entropy conservation properties of the tested non-dissipative schemes. Furthermore, the KEEP schemes are redefined based on the present entropy-error analysis. In the redefined KEEP schemes, the formulation does not contain logarithmic means, and the strictness of entropy conservation can be adjusted easily by the truncation order of the Maclaurin expansions. Finally, the present entropy-error analysis and the redefined KEEP schemes are extended to the generalized curvilinear coordinates.

High-order strongly nonlinear long wave approximation and solitary wave solution

We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.

An unstructured adaptive mesh refinement approach for computational fluid dynamics of reacting flows

A new Direct Numerical Simulation (DNS) code HAMISH with Adaptive Mesh Refinement (AMR) has been developed to simulate compressible reacting flow in a computationally economical manner. The focus is on problems where high gradients of temperature, density and species mass fraction remain localised, for example within the interior structure of flames. The resolution requirements of such local high-gradient regions often determine the global mesh spacing when uniform meshes are used. The numerical framework in HAMISH is based on an unstructured finite-volume approach together with a Runge-Kutta algorithm for time-stepping. The unstructured Cartesian mesh used in HAMISH allows for adaptive refinement and de-refinement of the local mesh depending on the demands of the flow physics and numerical accuracy. The code utilises an octree data structure and a Morton space-filling curve, allowing efficient cell division as well as straightforward parallel domain decomposition. The spatial discretisation scheme uses fourth-order polynomial reconstruction to evaluate the inter-cell fluxes. A third-order three-step explicit Runge-Kutta time-marching algorithm is used together with adaptive time-stepping with proportional-integral-type error control using an embedded Runge-Kutta scheme. The code is fully parallelised through domain decomposition over an arbitrary number of processors and exhibits a high level of parallel efficiency. In this paper the key capabilities of HAMISH are demonstrated based on a number of test cases. These test cases include (a) simulation of 1-D planar laminar premixed flames to demonstrate the capability of AMR in capturing the sharp gradients within the flame; (b) fully developed laminar channel flow, which shows that the code is capable of refining the mesh in the boundary layer next to the wall; (c) 2-D expanding flame under quiescent laminar condition, which demonstrates that the code is capable of dynamic local mesh refinement based on the position of the flame front; (d) 3-D non-reacting Taylor-Green vortex which demonstrates HAMISH can deal with vortical motion typical of turbulent flows; (e) 3-D premixed turbulent flame propagation under isotropic homogeneous decaying turbulence to demonstrate that the code can be employed for production level DNS of turbulent reacting flows.

Peridynamic Galerkin method: an attractive alternative to finite elements

This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker-delta property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.