Numerical Simulation of Compressible Vortical Flows Using a Conservative Unstructured-Grid Adaptive Scheme

A two-dimensional numerical scheme for the compressible Euler equations is presented and applied here to the simulation of exemplary compressible vortical flows. The proposed approach allows to perform computations on unstructured moving grids with adaptation, which is required to capture complex features of the flow-field. Grid adaptation is driven by suitable error indicators based on the Mach number and by element-quality constraints as well. At the new time level, the computational grid is obtained by a suitable combination of grid smoothing, edge-swapping, grid refinement and de-refinement. The grid modifications—including topology modification due to edge-swapping or the insertion/deletion of a new grid node—are interpreted at the flow solver level as continuous (in time) deformations of suitably-defined node-centered finite volumes. The solution over the new grid is obtained without explicitly resorting to interpolation techniques, since the definition of suitable interface velocities allows one to determine the new solution by simple integration of the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Numerical simulations of the steady oblique-shock problem, of the steady transonic flow and of the start-up unsteady flow around the NACA 0012 airfoil are presented to assess the scheme capabilities to describe these flows accurately.     

A Three-Level Multi-Continua Upscaling Method for Flow Problems in Fractured Porous Media

Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size one. Thus, multilevel methods are desirable for problems with complex heterogeneities and high contrast. In this paper, we propose a novel three-level upscaling method for flow problems in fractured porous media. Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct a nonlocal multi-continua upscaling (NLMC) method using an intermediate grid. The system resulting from NLMC gives solutions that have physical meaning. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of the Generalized Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final reduced model. We present simulation results for a two-dimensional model problem with a large number of fractures using the proposed three-level method.     

A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are defined by suitable level-set functions. The equations are discretized by classical nine-point stencil on interior points, while boundary conditions and high order reconstructions are used to define the field variables at ghost-points, which are grid nodes external to the domain with a neighbor inside the domain. The linear system arising from such discretization is solved by a multigrid strategy. The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources. The method is suitable to treat problems in which the geometry of the source often changes (explore the effects of different scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modified. Several numerical tests are successfully performed, which asses the effectiveness of the present approach.     

The Immersed Interface Method for Non-Smooth Rigid Objects in Incompressible Viscous Flows

In the immersed interface method, an object in a flow is formulated as a singular force, and jump conditions caused by the singular force are incorporated into numerical schemes to compute the flow. Previous development of the method considered only smooth objects. We here extend the method to handle non-smooth rigid objects with sharp corners in 2D incompressible viscous flows. We represent the boundary of an object as a polygonal curve moving through a fixed Cartesian grid. We compute necessary jump conditions to achieve boundary condition capturing on the object. We incorporate the jump conditions into finite difference schemes to solve the flow on the Cartesian grid. The accuracy, efficiency and robustness of our method are tested using canonical flow problems. The results demonstrate that the method has second-order accuracy for the velocity and first-order accuracy for the pressure in the infinity norm, and is extremely efficient and robust to simulate flows around non-smooth complex objects.     

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

The purpose of this article is to summarize our recent progress in high-order and high accurate CFD methods for flow problems with complex grids as well as to discuss the engineering prospects in using these methods. Despite the rapid development of high-order algorithms in CFD, the applications of high-order and high accurate methods on complex configurations are still limited. One of the main reasons which hinder the widely applications of these methods is the complexity of grids. Many aspects which can be neglected for low-order schemes must be treated carefully for high-order ones when the configurations are complex. In order to implement high-order finite difference schemes on complex multi-block grids, the geometric conservation law and block-interface conditions are discussed. A conservative metric method is applied to calculate the grid derivatives, and a characteristic-based interface condition is employed to fulfil high-order multi-block computing. The fifth-order WCNS-E-5 proposed by Deng [9, 10] is applied to simulate flows with complex grids, including a double-delta wing, a transonic airplane configuration, and a hypersonic X-38 configuration. The results in this paper and the references show pleasant prospects in engineering-oriented applications of high-order schemes.     

A Fast and Rigorously Parallel Surface Voxelization Technique for GPU-Accelerated CFD Simulations

This paper presents a fast surface voxelization technique for the mapping of tessellated triangular surface meshes to uniform and structured grids that provide a basis for CFD simulations with the lattice Boltzmann method (LBM). The core algorithm is optimized for massively parallel execution on graphics processing units (GPUs) and is based on a unique dissection of the inner body shell. This unique definition necessitates a topology based neighbor search as a preprocessing step, but also enables parallel implementation. More specifically, normal vectors of adjacent triangular tessellations are used to construct half-angles that clearly separate the per-triangle regions. For each triangle, the grid nodes inside the axis-aligned bounding box (AABB) are tested for their distance to the triangle in question and for certain well-defined relative angles. The performance of the presented grid generation procedure is superior to the performance of the GPU-accelerated flow field computations per time step which allows efficient fluid-structure interaction simulations, without noticeable performance loss due to the dynamic grid update.     

Decision tables and rule engines in organ allocation systems for optimal transparency and flexibility

Organ allocation systems have become complex and difficult to comprehend. We introduced decision tables to specify the rules of allocation systems for different organs. A rule engine with decision tables as input was tested for the Kidney Allocation System (ETKAS). We compared this rule engine with the currently used ETKAS by running 11 000 historical match runs and by running the rule engine in parallel with the ETKAS on our allocation system. Decision tables were easy to implement and successful in verifying correctness, completeness, and consistency. The outcomes of the 11 000 historical matches in the rule engine and the ETKAS were exactly the same. Running the rule engine simultaneously in parallel and in real time with the ETKAS also produced no differences. Specifying organ allocation rules in decision tables is already a great step forward in enhancing the clarity of the systems. Yet, using these tables as rule engine input for matches optimizes the flexibility, simplicity and clarity of the whole process, from specification to the performed matches, and in addition this new method allows well controlled simulations.     

An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems

In this paper, we study the numerical solution of singularly perturbed time-dependent convection-diffusion problems. To solve these problems, the backward Euler method is first applied to discretize the time derivative on a uniform mesh, and the classical upwind finite difference scheme is used to approximate the spatial derivative on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all temporal levels, we construct a positive monitor function, which is similar to the arc-length monitor function. Furthermore, the ε-uniform convergence of the fully discrete scheme is derived for the numerical solution. Finally, some numerical results are given to support our theoretical results.     

PVFMM: A Parallel Kernel Independent FMM for Particle and Volume Potentials

We describe our implementation of a parallel fast multipole method for evaluating potentials for discrete and continuous source distributions. The first requires summation over the source points and the second requiring integration over a continuous source density. Both problems require$\mathcal{O}$($N^2$) complexity when computed directly; however, can be accelerated to $\mathcal{O}$($N$) time using FMM. In our PVFMM software library, we use kernel independent FMM and this allows us to compute potentials for a wide range of elliptic kernels. Our method is high order, adaptive and scalable. In this paper, we discuss several algorithmic improvements and performance optimizations including cache locality, vectorization, shared memory parallelism and use of coprocessors. Our distributed memory implementation uses space-filling curve for partitioning data and a hypercube communication scheme. We present convergence results for Laplace, Stokes and Helmholtz (low wavenumber) kernels for both particle and volume FMM. We measure efficiency of our method in terms of CPU cycles per unknown for different accuracies and different kernels. We also demonstrate scalability of our implementation up to several thousand processor cores on the Stampede platform at the Texas Advanced Computing Center.     

A Two-Dimensional Second Order Conservative Front-Tracking Method with an Original Marker Advection Approach Based on Jump Relations

A two-dimensional front-tracking method is developed for handling complex shape interfaces satisfying the volume conservation. In order to validate the proposed front-tracking method, a complete convergence study is carried out on several analytical test cases for which the interface is widely stretched and deformed. Comparisons to different existing approaches show that our front-tracking method is second order accurate in space with lower errors than existing interface tracking techniques of the literature.We also propose an original marker advection method which takes into account the jump relations valid at interface in order to deal with the contrast of physical properties encountered in two-phase flow simulations. The conservative front-tracking method computed in this work is shown to be able to describe interfaces with high accuracy even for poorly resolved Eulerian grids.     

A Local Velocity Grid Approach for BGK Equation

The solution of complex rarefied flows with the BGK equation and the Discrete Velocity Method (DVM) requires a large number of velocity grid points leading to significant computational costs. We propose an adaptive velocity grid approach exploiting the fact that locally in space, the distribution function is supported only by a sub-set of the global velocity grid. The velocity grid is adapted thanks to criteria based on local temperature, velocity and on the enforcement of mass conservation. Simulations in 1D and 2D are presented for different Knudsen numbers and compared to a global velocity grid BGK solution, showing the computational gain of the proposed approach.     

Performance of different optimal charging schemes in a solar charging station using dynamic programming

Electric Vehicles (EVs) are gradually replacing conventional vehicles as they are environmentally friendly and cause less pollution problems. Unregulated charging has severe impacts on the distribution grid and may incur EV owners higher charging costs. Therefore, controlled charging infrastructures to supply the charging needs of large numbers of EVs are of vital importance. In this article, an optimal control scenario is presented to formulate the charge scheduling problem of EVs in a solar charging station (CS). Two different objective functions are considered. The first objective function holds for minimizing the total charging cost of EVs. In this case, the benefits of Vehicle-to-Grid (V2G) are investigated by comparing the charging costs of EVs with and without this capability. The total EV charging costs and grid benefits are also investigated in the second objective function which holds for minimizing the extracted power from the grid. A modified version of Dynamic Programming is used to solve the large state-space model defined for the optimal control problem with extremely shorter computation time and minimal loss of optimality. Extensive simulations are done in two representative summer and winter climates to determine the role of solar energy in the CS performance. The results show that in the cost minimization algorithms, significant savings for EV owners and a smooth load shape for the grid are achieved. For the minimized power from the grid algorithm, a total near Photovoltaic (PV)-curve charging power is obtained to exploit the PV power as much as possible to minimize the impacts on the grid.     

Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids. The generalized Navier boundary condition proposed by Qian et al. [1] is adopted here. We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time. We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid. We also derive the discrete energy law for the droplet displacement case, which is slightly different due to the boundary conditions. The accuracy and stability of the scheme are validated by examples, results and estimate order.     

An alternating direction method of multipliers algorithm for symmetric model predictive control

This article presents an alternating direction method of multipliers (ADMM) algorithm for solving large‐scale model predictive control (MPC) problems that are invariant under the symmetric‐group. Symmetry was used to find transformations of the inputs, states, and constraints of the MPC problem that decompose the dynamics and cost. We prove an important property of the symmetric decomposition for the symmetric‐group that allows us to efficiently transform between the original and decomposed symmetric domains. This allows us to solve different subproblems of a baseline ADMM algorithm in different domains where the computations are less expensive. This reduces the computational cost of each iteration from quadratic to linear in the number of repetitions in the system. In addition, we show that the memory complexity for our ADMM algorithm is also linear in number of repetitions in the system, rather than the typical quadratic complexity. We demonstrate our algorithm for two case studies; battery balancing and heating, ventilation, and air conditioning. In both case studies, the symmetric algorithm reduced the computation‐time from minutes to seconds and memory usage from tens of megabytes to tens or hundreds of kilobytes, allowing the previously nonviable MPCs to be implemented in real time on embedded computers with limited computational and memory resources.     

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.     

Transition of Liesegang Precipitation Systems: Simulations with an Adaptive Grid PDE Method

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.     

Rabbit as a distraction model—pitfalls

The principles of distraction osteogenesis have been successfully applied to the craniofacial skeleton of different animals. The rabbit, in particular, has been evaluated as a model by enumerable authors. To our knowledge, however, none of the studies either report the causes of premature euthanization or the pitfalls leading to the untimely death of the animal. We here describe our experience with 30 rabbits used as a model for mandibular distraction osteogenesis and suggest precautions to take in order to avoid unforeseen problems. Thirty skeletally mature New Zealand white rabbits were used. Fifteen animals had bilateral distraction devices placed on the anterior mandible, and another 15 underwent unilateral distraction osteogenesis. In both groups, 12 animals were euthanized prematurely due to complications that included excessive weight loss (malnutrition), anesthesia/animal-related problems, and distraction device failure. The remaining 18 animals tolerated the operative procedure well. Indisputably, rabbit is an excellent choice for craniofacial experiments, but because of its complex anatomy and physiology, an unexpected outcome frequently occurs. We believe that the following suggestions in relation to the pre-operative selection of a suitable animal model, operative technique, and management of eating problems may help the researcher to choose an appropriate animal and avoid complications leading to early death.     

Numerical Path Integral Approach to Quantum Dynamics and Stationary Quantum States

Applicability of Feynman path integral approach to numerical simulations of quantum dynamics of an electron in real time domain is examined. Coherent quantum dynamics is demonstrated with one dimensional test cases (quantum dot models) and performance of the Trotter kernel as compared with the exact kernels is tested. Also, a novel approach for finding the ground state and other stationary sates is presented. This is based on the incoherent propagation in real time. For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling. We asses the numerical prerequisites for all of the above.     

Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems

In this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms. Classic two-mode approximation method and direct numerical minimization are applied to the model. In the latter approach, we apply the projection method to deal with the potentially quasiperiodic ground states. A variable cell method of optimizing the shape and size of higher-dimensional periodic cell is developed to minimize the free energy with respect to the order parameters. Based on the developed numerical methods, we rediscover decagonal and dodecagonal quasicrystalline phases, and find diverse periodic phases and complex modulated phases. Furthermore, phase diagrams are obtained in various phase spaces by comparing the free energies of different candidate structures. It does show not only the important roles of system parameters, but also the effect of optimizing computational domain. In particular, the optimization of computational cell allows us to capture the ground states and phase behavior with higher fidelity. We also make some discussions on our results and show the potential of applying our numerical methods to a larger class of mean-field free energy functionals.     

A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.