A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data

In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS) method, which provides a numerical approximation of the Dirichlet-to-Neumann (DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the boundary of the domain and does not require a structured mesh, and only needs a covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is introduced for better error controls. The effect of errors from the Feynman-Kac formula based path integral WOS method on the overall accuracy of the BIE-WOS method is analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs. For the special case of flat patches, it is shown that the second kind integral equation of BIE-WOS method can be simplified where the local BIE solutions can be given in closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of the boundary, resulting in a complete independent computation using a large number of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN mapping of exterior Laplace problems with Dirichlet data for several domains.     

Explicit Computation of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators

The Optimized Schwarz Waveform Relaxation algorithm, a domain decomposition method based on Robin transmission condition, is becoming a popular computational method for solving evolution partial differential equations in parallel. Along with well-posedness, it offers a good balance between convergence rate, efficient computational complexity and simplicity of the implementation. The fundamental question is the selection of the Robin parameter to optimize the convergence of the algorithm. In this paper, we propose an approach to explicitly estimate the Robin parameter which is based on the approximation of the transmission operators at the subdomain interfaces, for the linear/nonlinear Schrödinger equation. Some illustrating numerical experiments are proposed for the one- and two-dimensional problems.     

A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes

In this paper, a conservative parallel iteration scheme is constructed to solve nonlinear diffusion equations on unstructured polygonal meshes. The design is based on two main ingredients: the first is that the parallelized domain decomposition is embedded into the nonlinear iteration; the second is that prediction and correction steps are applied at subdomain interfaces in the parallelized domain decomposition method. A new prediction approach is proposed to obtain an efficient conservative parallel finite volume scheme. The numerical experiments show that our parallel scheme is second-order accurate, unconditionally stable, conservative and has linear parallel speed-up.     

Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.     

A Wavelet-Adaptive Method for Multiscale Simulation of Turbulent Flows in Flying Insects

We present a wavelet-based adaptive method for computing 3D multiscale flows in complex, time-dependent geometries, implemented on massively parallel computers. While our focus is on simulations of flapping insects, it can be used for other flow problems. We model the incompressible fluid with an artificial compressibility approach in order to avoid solving elliptical problems. No-slip and in/outflow boundary conditions are imposed using volume penalization. The governing equations are discretized on a locally uniform Cartesian grid with centered finite differences, and integrated in time with a Runge–Kutta scheme, both of 4th order. The domain is partitioned into cubic blocks with different resolution and, for each block, biorthogonal interpolating wavelets are used as refinement indicators and prediction operators. Thresholding the wavelet coefficients allows to generate dynamically evolving grids, and an adaption strategy tracks the solution in both space and scale. Blocks are distributed among MPI processes and the grid topology is encoded using a tree-like data structure. Analyzing the different physical and numerical parameters allows us to balance their errors and thus ensures optimal convergence while minimizing computational effort. Different validation tests score accuracy and performance of our new open source code, WABBIT. Flow simulations of flapping insects demonstrate its applicability to complex, bio-inspired problems.     

Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

In order to solve the partial differential equations that arise in the Hartree-Fock theory for diatomic molecules and in molecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variable model problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9-point difference formula produces the same level of accuracy with fewer points. The domain decomposition method has the strength that it can be applied to problems with a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments thatare calculated independently. Since the domain decomposition approach makes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.     

Lumped Parameter Outflow Models for 1-D Blood Flow Simulations: Effect on Pulse Waves and Parameter Estimation

Several lumped parameter, or zero-dimensional (0-D), models of the microcirculation are coupled in the time domain to the nonlinear, one-dimensional (1-D) equations of blood flow in large arteries. A linear analysis of the coupled system, together with in vivo observations, shows that: (i) an inflow resistance that matches the characteristic impedance of the terminal arteries is required to avoid non-physiological wave reflections; (ii) periodic mean pressures and flow distributions in large arteries depend on arterial and peripheral resistances, but not on the compliances and inertias of the system, which only affect instantaneous pressure and flow waveforms; (iii) peripheral inertias have a minor effect on pulse waveforms under normal conditions; and (iv) the time constant of the diastolic pressure decay is the same in any 1-D model artery, if viscous dissipation can be neglected in these arteries, and it depends on all the peripheral compliances and resistances of the system. Following this analysis, we propose an algorithm to accurately estimate peripheral resistances and compliances from in vivo data. This algorithm is verified against numerical data simulated using a 1-D model network of the 55 largest human arteries, in which the parameters of the peripheral windkessel outflow models are known a priori. Pressure and flow waveforms in the aorta and the first generation of bifurcations are reproduced with relative root-mean-square errors smaller than 3%.     

Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.     

Gyrofluid Simulation of Ion-Scale Turbulence in Tokamak Plasmas

An improved three-field gyrofluid model is proposed to numerically simulate ion-scale turbulence in tokamak plasmas, which includes the nonlinear evolution of perturbed electrostatic potential, parallel ion velocity and ion pressure with adiabatic electron response. It is benchmarked through advancing a gyrofluid toroidal global (GFT_G) code as well as the local version (GFT_L), with the emphasis of the collisionless damping of zonal flows. The nonlinear equations are solved by using Fourier decomposition in poloidal and toroidal directions and semi-implicit finite difference method along radial direction. The numerical implementation is briefly explained, especially on the periodic boundary condition in GFT_L version. As a numerical test and also practical application, the nonlinear excitation of geodesic acoustic mode (GAM), as well as its radial structure, is investigated in tokamak plasma turbulence.     

A NURBS-Enhanced Finite Volume Method for Steady Euler Equations with Goal-Oriented $h$-Adaptivity

In [A NURBS-enhanced finite volume solver for steady Euler equations, X. C. Meng, G. H. Hu, J. Comput. Phys., Vol. 359, pp. 77–92], a NURBS-enhanced finite volume method was developed to solve the steady Euler equations, in which the desired high order numerical accuracy was obtained for the equations imposed in the domain with a curved boundary. In this paper, the method is significantly improved in the following ways: (i) a simple and efficient point inversion technique is designed to compute the parameter values of points lying on a NURBS curve, (ii) with this new point inversion technique, the $h$-adaptive NURBS-enhanced finite volume method is introduced for the steady Euler equations in a complex domain, and (iii) a goal-oriented a posteriori error indicator is designed to further improve the efficiency of the algorithm towards accurately calculating a given quantity of interest. Numerical results obtained from a variety of numerical experiments with different flow configurations successfully show the effectiveness and robustness of the proposed method.     

An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem.     

Weak Galerkin Method for Second-Order Elliptic Equations with Newton Boundary Condition

The weak Galerkin (WG) method is a nonconforming numerical method for solving partial differential equations. In this paper, we introduce the WG method for elliptic equations with Newton boundary condition in bounded domains. The Newton boundary condition is a nonlinear boundary condition arising from science and engineering applications. We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions. The error estimates are derived. Numerical experiments are presented to verify the theoretical analysis.     

A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.     

Shape optimization analysis: First- and second-order necessary conditions

In this paper, first- and second-order necessary conditions for optimality are studied for a domain optimization problem. The optimization problem considered is the minimization of an objective function defined on the domain boundary through the solution of a boundary value problem. In order to derive the first and second variations of the objective function due to boundary variation, the first and second variations of the solution of the boundary value problem are calculated using a perturbation technique. An iterative shape optimization algorithm for potential flow problems in R² with Dirichlet boundary conditions is presented. In the algorithm a boundary element method (BEM) is employed to solve the Laplace equation numerically. The validity and accuracy of the algorithm have been verified on a problem where the final solution is known. Finally, the problem of designing a 90° bend for two-dimensional potential flow is solved.     

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.     

Numerical Simulation of Glottal Flow in Interaction with Self Oscillating Vocal Folds: Comparison of Finite Element Approximation with a Simplified Model

In this paper the numerical method for solution of an aeroelastic model describing the interactions of air flow with vocal folds is described. The flow is modelled by the incompressible Navier-Stokes equations spatially discretized with the aid of the stabilized finite element method. The motion of the computational domain is treated with the aid of the Arbitrary Lagrangian Eulerian method. The structure dynamics is replaced by a mechanically equivalent system with the two degrees of freedom governed by a system of ordinary differential equations and discretized in time with the aid of an implicit multistep method and strongly coupled with the flow model. The influence of inlet/outlet boundary conditions is studied and the numerical analysis is performed and compared to the related results from literature.     

Boundary Control Problems in Convective Heat Transfer with Lifting Function Approach and Multigrid Vanka-Type Solvers

This paper deals with boundary optimal control problems for the heat and Navier-Stokes equations and addresses the issue of defining controls in function spaces which are naturally associated with the volume variables by trace restriction. For this reason we reformulate the boundary optimal control problem into a distributed problem through a lifting function approach. The stronger regularity requirements which are imposed by standard boundary control approaches can then be avoided. Furthermore, we propose a new numerical strategy that allows solving the coupled optimality system in a robust way for a large number of unknowns. The optimality system resulting from a finite element discretization is solved by a local multigrid algorithm with domain decomposition Vanka-type smoothers. The purpose of these smoothers is to solve the optimality system implicitly over subdomains with a small number of degrees of freedom, in order to achieve robustness with respect to the regularization parameters in the cost functional. We present the results of some test cases where temperature is the observed quantity and the control quantity corresponds to the boundary values of the fluid temperature in a portion of the boundary. The control region for the observed quantity is a part of the domain where it is interesting to match a desired temperature value.     

A Scalable Domain Decomposition Method for Ultra-Parallel Arterial Flow Simulations

Ultra-parallel flow simulations on hundreds of thousands of processors require new multi-level domain decomposition methods. Here we present such a new two-level method that has features both of discontinuous and continuous Galerkin formulations. Specifically, at the coarse level the domain is subdivided into several big patches and within each patch a spectral element discretization (fine level) is employed. New interface conditions for the Navier-Stokes equations are developed to connect the patches, relaxing the C⁰continuity and minimizing data transfer at the patch interface. We perform several 3D flow simulations of a benchmark problem and of arterial flows to evaluate the performance of the new method and investigate its accuracy.     

Computational fluid dynamics and vascular access

Anastomotic intimal hyperplasia caused by unphysiological hemodynamics is generally accepted as a reason for dialysis access graft occlusion. Optimizing the venous anastomosis can improve the patency rate of arteriovenous grafts. The purpose of this study was to examine, evaluate, and characterize the local hemodynamics and, in particular, the wall shear stresses in conventional venous end-to-side anastomosis and in patch form anastomosis (Venaflo) by three-dimensional computational fluid dynamics (CFD). We investigated the conventional form of end-to-side anastomosis and a new patch form by numerical simulation of blood flow. The numerical simulation was done with a finite volume-based algorithm. The anastomotic forms were constructed with usual size and fixed walls. Subdividing the flow domain into multiple control volumes solved the fundamental equations. The boundary conditions were identical for both forms. The velocity profile of the patch form is better than that for the conventional form. The region of high static pressure caused by flow stagnation is reduced on the vein floor. The anastomotic wall shear stress is decreased. The results of this study strongly support patch form use to reduce the incidence of intimal hyperplasia and venous anastomotic stenoses.     

Numerical Simulation of Airfoil Vibrations Induced by Turbulent Flow

The subject of the paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil with large amplitudes. The airfoil with three degrees of freedom performs rotation around an elastic axis, oscillations in the vertical direction and rotation of a flap. The numerical simulation consists of the finite element solution of the Reynolds averaged Navier-Stokes equations combined with Spalart-Allmaras or k−ω turbulence models, coupled with a system of nonlinear ordinary differential equations describing the airfoil motion with consideration of large amplitudes. The time-dependent computational domain and approximation on a moving grid are treated by the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Due to large values of the involved Reynolds numbers an application of a suitable stabilization of the finite element discretization is employed. The developed method is used for the computation of flow-induced oscillations of the airfoil near the flutter instability, when the displacements of the airfoil are large, up to ±40 degrees in rotation. The paper contains the comparison of the numerical results obtained by both turbulence models.