Spatial Discretization Schemes Research Articles

A quantum computing concept for 1-D elastic wave simulation with exponential speedup

Summary Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real quantum computer. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based quantum computer. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real quantum computer IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyze the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wave field simulations that run exponentially faster than simulations on classical computers.

Development and benchmarking of space-time kinetics solver in REDAC

A space–time kinetics solver based on finite volume approach has been developed for REactor Dynamics Analysis Code (REDAC). It solves 3D multi-group neutron diffusion equation, coupled with the heat transport equations. Code offers flexibility in terms of reactor geometry, boundary conditions and discretization. Efficient numerical methods have been employed for solving the governing equations. The performance of the code has been demonstrated by solving diverse benchmark problems. It is shown that REDAC is able to predict important static and dynamic core parameters with good accuracy. The framework of REDAC easily lends itself for sensitivity studies with respect to spatial discretization and temporal integration schemes. From these studies, it is observed that spatial resolution could have significant influence on accuracy and the semi-implicit method performs better than the implicit method. Improved quasi-static method requires less computational time compared to the direct method, however, its accuracy depends on the time step size.

Energy-Dependent, Self-Adaptive Mesh h(p)-Refinement of an Interior-Penalty Scheme for a Discontinuous Galerkin Isogeometric Analysis Spatial Discretization of the Multi-Group Neutron Diffusion Equation with Dual-Weighted Residual Error Measures

Energy-dependent self-adaptive mesh refinement algorithms are developed for a symmetric interior-penalty scheme for a discontinuous Galerkin spatial discretization of the multi-group neutron diffusion equation using NURBS-based isogeometric analysis (IGA). The spatially self-adaptive algorithms employ both mesh (h) and polynomial degree (p) refinement. The discretized system becomes increasingly ill-conditioned for increasingly large penalty parameters; and there is no gain in accuracy for over penalization. Therefore, optimized penalty parameters are rigorously calculated, for general element types, from a coercivity analysis of the bilinear form. Local mesh refinement allows for a better allocation of computational resources; and thus, more accuracy per degree of freedom. Two a posteriori interpolation-based error measures are proposed. The first heuristically minimizes local contributions to the discretization error, which becomes competitive for global quantities of interest (QoIs). However, for localized QoIs, over energy-dependent meshes, certain multi-group components may become under-resolved. The second employs duality arguments to minimize important error contributions, which consistently and reliably reduces the error in the QoI.

A quantum algorithm for computing dispersal of submarine volcanic tephra

In this paper, we develop a quantum computing algorithm for solving the partial differential equation (PDE) for tephra dispersal through advection in the semi-infinite horizontal buoyant region of a submarine volcanic eruption. The concentration of pyroclastic particles in the fluid domain of a hydrothermal megaplume provides important information about the rate of volcanic energy release, mechanism of formation of the megaplume, and submarine depositional patterns. This work leveraging on previous works [F. Gaitan, NPJ Quantum Inf. 6, 61 (2020); F. Gaitan, Adv. Quantum Tech. 4, 2100055 (2021)] further opens up opportunities to solve wider classes of PDEs with different applications of interest. Some additional specific contributions of this work are transforming the semi-infinite spatial domain problem into a problem on a finite spatial domain for applying the quantum algorithm, and the investigation into the effect of spatial and temporal resolution on the solution of PDEs for the quantum algorithm. Furthermore, possible modification of the algorithm with different spatial discretization schemes has been presented and their influence and implications on the solution of the PDE have been discussed. Also, studies are conducted to examine the effect of regularity conditions in time and the presence of statistical noise in the spatial domain, on the solutions obtained using quantum algorithms. The study in this paper paves an important pathway to venture into other types of advection-diffusion problems.

Numerical discretization for Fisher-Kolmogorov problem with nonlocal diffusion based on mixed Galerkin BDF2 scheme

Nonlocal problems involving fourth-order terms pose several difficulties such as numerical discretization and its related convergences analysis. In this paper, the well-posedness of the extended Fisher-Kolmogorov equation with nonlocal diffusion is first analyzed using the Faedo-Galerkin technique and the classical compactness arguments. Moreover, we adopt a BDF2 scheme for time discretization and a mixed Galerkin scheme for spatial discretization. Then, we derive the optimal order convergence rates of the fully discrete system. Finally, some numerical simulations and convergence results are provided to confirm the theoretical results and the accuracy of the proposed scheme.

Design and Application of DG-FEM Basis Functions for Neutron Transport on Two-Dimensional and Three-Dimensional Hexagonal Meshes

Reactor design requires safety studies to ensure that the reactors will behave appropriately under incidental or accidental situations. Safety studies often involve multiphysics simulations where several branches of reactor physics are necessary to model a given phenomenon. In those situations, it has been observed that the neutron transport part is still a bottleneck in terms of computational times, with more than 80% of the total time. In the case of hexagonal lattice reactors, transport solvers usually invert the discretised Boltzmann equation by discretising the regular hexagon into lozenges or triangles. In this work, we seek to reduce the computational burden of the neutron transport solver by designing a numerical spatial discretisation scheme that would be more appropriate for honeycomb meshes. In our past research efforts, we have set up interesting discretisation schemes in the finite element setting in 2D, and we wish to extend them to 3D geometries that are prisms with a hexagonal base. In 3D, a rigorous method was derived to shrink the tensor product between 2D and 1D bases to minimum terms. We have applied these functions successfully on a reactor benchmark—Takeda Model 4—to compare and contrast the numerical results in a physical setting.

Numerical Study of the Effect of Primary Nozzle Geometry on Supersonic Gas-Solid Jet of Bypass Injected Dry Powder Fire Extinguishing Device

A two-way coupled model between polydisperse particle phases with compressible gases and a density-based coupling implicit solution method, combining the third-order MUSCL with QUICK spatial discretization scheme and the second-order temporal discretization scheme, are constructed based on the discrete-phase model (DPM) and the stochastic wander model (DRWM) in the Eulerian–Lagrangian framework in conjunction with a unitary particulate source (PSIC) approach and the SST k-ω turbulence model. The accuracy of the numerical prediction method is verified using previous supersonic nozzle gas-solid two-phase flow experiments. Numerical simulation of a two-phase jet of dry powder extinguishing agent gas with pilot-type supersonic nozzle was performed to analyze the influence of geometrical parameters, such as the length ratio rL and the area ratio rA of the main nozzle on the two-phase flow field, as well as on the jet performance indexes, such as the particle mean velocity vp,a, velocity inhomogeneity Φvp, particle dispersion Ψp, particle mean acceleration ap,a, etc. By analyzing the parameters, we indicate the requirements for the combination of jet performance metrics for different flame types such as penetrating, spreading, and dispersing.

Pricing a resettable convertible bond based on decomposition method and PDE models

In this paper, a partial differential equation approach based on the underlying stock price path decomposition is developed to price an American-style resettable convertible bond. The American-style resettable convertible bond is viewed as a mixture of three simple securities, which can be used to replicate the feature of payoffs of the resettable convertible bond completely. The partial differential equations under the Black–Scholes framework are established to price these simple securities. An implicit Euler method is used to discretize the first-order time derivative while a central finite difference method on a piecewise uniform mesh is used to discretize the spatial derivatives. The error estimates are developed by using the maximum principle in two mesh sets both for the time semi-discretization scheme and the spatial discretization scheme, respectively. It is proved that the scheme is first-order convergent for the time variable and second-order convergent for the spatial variable. Numerical experiments support these theoretical results.

Spatial Discretization for Stochastic Semilinear Superdiffusion Driven by Fractionally Integrated Multiplicative Space–Time White Noise

We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in the Caputo sense with an order α∈ (1, 2). A spatial discretization scheme is introduced by approximating the space–time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied, and the optimal error estimates that depend on the smoothness of the initial values are established.

Development of a Fully-Implicit Second-Order system Thermal-Hydraulics analysis code for reactor modeling and simulations

There has been continuous advancement in numerical techniques and software engineering in the field of system thermal-hydraulics over the past decade. The current state-of-the-art involves the use of fully-implicit high-resolution schemes in conjunction with the Jacobian-Free Newton-Krylov (JFNK) method for both single-phase and two-phase flows. However, the application of these advanced schemes in system thermal-hydraulics codes is not a common practice. Given the opportunity and the need to develop a new modern system thermal–hydraulic code from scratch, there is no doubt that these recent advancements should be leveraged. In this work, we introduce a new modern object-oriented system thermal-hydraulics code named RETA. RETA is characterized by several key features, including: second-order temporal and spatial discretization schemes for both incompressible and compressible flows, as well as single-phase and two-phase flows; fully-implicit solution schemes utilizing both the Newton and PJFNK methods; and an object-oriented design of major fluid components and physical models, which allows for easy extension. A series of verification and demonstration studies are conducted to verify the implementation, demonstrate the accuracy improvement, and evaluate the performance of the new code. It is concluded that results from the second-order scheme are generally more reliable. Without the concern of a code stability issue, the second-order scheme should be the preferred and default choice for a code user or an analyst.

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.

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.

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.