Treatment Of Boundary Conditions Research Articles

Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms \textit{both biased and unbiased} simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory. Note: this is an earlier version of the work Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations.

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

Some comments on using fractional derivative operators in modeling non-local diffusion processes

We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. The numerical properties of the scheme resulting from the flux laws are analyzed. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann–Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.

Penalty function method for imposing nonlinear multi freedom and multi node constraints in finite element analysis of frame systems

This paper focuses on the treatment of nonlinear multi-freedom and multi-point boundary condition in finite element analysis of frame systems. The treatment of boundary constraints is required to produce modified system of equations based on master stiffness equations considering nonlinear constraints. For imposing nonlinear multi freedom and multi point constraints, the Penalty Augmentation method and Lagrange Multiplier Adjunction method are better in many applications. In present paper, the Penalty Augmentation method is using for implementation of nonlinear boundary constraints. The nonlinear relationship considerably increases the difficulty in constructing and solving the modified system of equations. The Newton Raphson method, have been the most powerful and popular method for solution of nonlinear equations, is used for solving studied problem. Using the Newton Raphson technique, this paper develops the incremental-iterative algorithm to solve the nonlinear modified system of equations. Based on the presented algorithm, the paper proposed calculation procedure and established programs for determining internal forces and displacements of frames having nonlinear multi freedom and multi point constrains boundary. The numerical test results using the proposed method show the efficiency and reliability of proposed algorithm.

Nonreflecting Boundary Conditions for a CSF Model of Fourth Ventricle: Spinal SAS Dynamics.

In this paper, we introduce a one-dimensional model for analyzing the cerebrospinal fluid dynamics within the fourth ventricle and the spinal subarachnoid space (SSAS). The model has been derived starting from an original model of Linninger et al. and from the detailed mathematical analysis of two different reformulations. We show the steps of the modelization and the rigorous analysis of the first-order nonlinear hyperbolic system of equations which rules the new CSF model, whose conservative-law form and characteristic form are required for the boundary conditions treatment. By assuming sub-critical flows, for the particular dynamics we are dealing with, the most desirable option is to employ the nonreflecting boundary conditions, that allow the simple wave associated with the outgoing characteristic to exit the computational domain with no reflections. Finally, we carry out some numerical simulations related to different cerebral physiological conditions.

An accurate treatment of non-homogeneous boundary conditions for development of the BEM

This paper proposes an enhancement of the treatment of non-homogeneous boundary conditions to improve the boundary element method (BEM) formulation. The standard formulation is modified by introducing the boundary conditions in the integral kernels. The boundary conditions are implicitly defined through known parameters depending on the geometry, rather than by prescribing nodal values as is done in the standard formulation. The main advantage of this procedure is that the right-hand side of the system of equations is integrated taking the exact distribution of loads into account. This approach is implemented in the Bézier–Bernstein space to yield a geometry-independent field approximation. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The application of the proposed method covers the resolution of complex boundary value problems as optimization with uncertain data, material modelling with graded impedance, and the definition of general boundary constraints. The performance of the proposed method is shown by solving the Helmholtz equation in two dimensions. The proposed method is numerically compared to the standard BEM formulation in two benchmark problems. Finally, the application of complex impedance boundary conditions is analysed in a numerical example.

On the verification of CFD solvers of all orders of accuracy on curved wall-bounded domains and for realistic RANS flows

This paper aims at extending a recent high-order code verification methodology to curved wall-bounded domains as well as to realistic turbulent flows such as the flat plate boundary layer modelled by the Reynolds-averaged Navier-Stokes (RANS) equations. Two new manufactured solutions (MSs) are introduced with demonstrated ability to verify the treatment of slip and no-slip boundary conditions in high-order frameworks on curved domains. These MSs serve as well to discuss the impact of the method of computation of boundary normals on the order of accuracy (OOA) of the solution at the wall. Furthermore, two turbulent boundary layer MSs from literature, devised to mimic the genuine features of RANS-modelled flows in the vicinity of wall, are compared in terms of their suitability to achieving high-order accuracy. A number of useful concepts in verification are explored through these cases such as the limit values of the Spalart-Allmaras (SA) turbulence model source terms at the wall, the verification of the modified vorticity term of the negative SA model, the grid sensitivity of wall-bounded turbulent flows, the inadequacy of substituting solution verification to code verification and the effect of non-dimensionalization of the solution on the minimization of iterative errors via residual convergence. In all cases, demonstrations are carried for orders of accuracy up to the sixth.

A gPAV-based unconditionally energy-stable scheme for incompressible flows with outflow/open boundaries

We present an unconditionally energy-stable scheme for approximating the incompressible Navier–Stokes equations on domains with outflow/open boundaries. The scheme combines the generalized Positive Auxiliary Variable (gPAV) approach and a rotational velocity-correction type strategy, and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions. The discrete energy stability of the proposed scheme has been proven, irrespective of the time step sizes. Within each time step the scheme entails the computation of two velocity fields and two pressure fields, by solving an individual de-coupled Helmholtz (including Poisson) type equation with a constant pre-computable coefficient matrix for each of these field variables. The auxiliary variable, being a scalar number, is given by a well-defined explicit formula within a time step, which ensures the positivity of its computed values. Extensive numerical experiments with several flows involving outflow/open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes.

Variational formulation of plasma dynamics

Hamilton's principle is applied to obtain the equations of motion for fully relativistic collision-free plasma. The variational treatment is presented in both the Eulerian and Lagrangian frameworks. A Clebsch representation of the plasma fluid equations shows the connection between the Lagrangian and Eulerian formulations, clarifying the meaning of the multiplier in Lin's constraint. The existence of a fully relativistic hydromagnetic Cauchy invariant is demonstrated. The Lagrangian approach allows a straightforward determination of the Hamiltonian density and energy integral. The definitions of momentum, stress, and energy densities allow one to write the conservation equations in a compact and covariant form. The conservation equations are also written in an integral form with an emphasis on a generalized virial theorem. The treatment of boundary conditions produces a general expression for energy density distribution in plasma fluid.

A response factor approach to modelling long-term thermal behaviour of energy piles

Abstract Predicting the long-term thermal performance of an energy pile ground heat exchanger system requires a computationally efficient model that captures the three-dimensional features and interacting thermal boundary conditions found in such systems. We present a model that employs a response factor approach to predicting long-term thermal behaviour based on a Dynamic Thermal Network representation. This allows fully three-dimensional geometric representation of the problem and application of boundary conditions at the ground and building sub-surfaces in addition to the pipes. The model has been validated using data from long-term monitoring of a pile in Belgium excited by a combination of heating and cooling demands in a range of environmental conditions. We further demonstrate application of the model to simulated annual building energy demands and the significance of treatment of the upper surface boundary conditions. The model is shown to be capable of representing building system and environmental conditions effectively and furthermore computationally efficient enough for routine design and simulation tasks.

Very high-order method on immersed curved domains for finite difference schemes with regular Cartesian grids

A new very high-order technique for solving conservation laws with curved boundary domains is proposed. A Finite Difference scheme on Cartesian grids is coupled with an original ghost cell method that provide accurate approximations for smooth solutions. The technology is based on a specific least square method with restrictions that enables to handle general Robin conditions. Several examples in two-dimensional geometries are presented for the unsteady Convection–Diffusion equation and the Euler equations. A fifth-order WENO scheme is employed with matching fifth-order reconstruction at the boundaries. Arbitrary high-order reconstruction for smooth flows is achievable independently of the underlying differential equation since the method works as a black-box dedicated to boundary condition treatment.

Efficient Design Optimization of Acoustic Liners for Engine Noise Reduction

Engine noise reduction using acoustic liner panels is an established technology in the design of bypass turbofan engines. Using acoustic impedance modeling techniques in aeroacoustic simulations, quick assessment of different liner configurations can be made. To minimize the engine noise in the best possible way, optimal liner configurations are ideally determined by employing numerical optimization methods. In general, to exhaust the full potential of the lining technology, it is desired to use a large number of design parameters in the optimization process. In the present work, a multilevel optimization framework is presented for the design of acoustic liner panels, which can efficiently handle large design vectors. The multilevel method combines a global search strategy with a local gradient-based optimization method. The discrete adjoint solver, which is used to evaluate the liner sensitivities required in the local search, has been developed using algorithmic differentiation techniques. This approach allows consistent treatment of boundary conditions in the adjoint part, thereby increasing the robustness and accuracy of the adjoint solver. In this way, the discrete adjoint solver fully incorporates the features of the underlying aeroacoustic solver, such as the impedance modeling and nonreflecting boundary conditions. The feasibility and the efficiency of the multilevel optimization strategy are demonstrated by finding the optimal liner parameters in a turbofan engine bypass duct configuration.

A fast and efficient integration of boundary conditions into a unified CUDA Kernel for a shallow water solver lattice Boltzmann Method

In this work, we present an exhaustive performance analysis of the integration of boundary conditions in a unified CUDA kernel for a lattice Boltzmann shallow water solver. This kernel is implemented under the pull scheme approach of the lattice Boltzmann method. The analysis is performed simulating open ocean domains with open and bounce-back boundary conditions. Boundary conditions treatment is divided in two steps: identification of the classes of the distribution function components in a node and branching handling. Several methods are proposed for each step, and all the combinations of them are tested with different hardware, domain size and floating point precision. Results show that high performance is achieved when using two binary precomputed values for class identification, while handling branchings with Boolean multiplication should be avoided. A full report of the MLUPS (Millions of Lattice Updates Per Second) ratio achieved with each test is presented.

Spectral analysis of guided wave propagation in discretized domains under local interactions

The paper presents a spectral analysis of wave propagation in bounded two-dimensional elastic media discretized under assumptions of two selected local computational strategies. The Local Interaction Simulation Approach and Cellular Automata for Elastodynamics are first overviewed with particular emphasis on their fundamental assumptions, treatment of boundary conditions, and resulting formulations. Each method’s numerical dispersion characteristics for bulk waves are then analyzed using a closed set of iteration equations, after which guided Lamb wave dispersion characteristics are analyzed for homogeneous and layered plates. In all cases, comparisons to finite element dispersion characteristics, re-formulated in the form of local iteration equations, provide insight into the relative advantages and disadvantages of the local formulations and reveal non-physical effects induced by numerical discretization. Finally, verification of the analytical dispersion results is presented using transient simulations with accompanying spectral analysis.

Assessment of Smoothed Particle Hydrodynamics (SPH) models for predicting wall heat transfer rate at complex boundary

Nowadays, the use of Smoothed Particle Hydrodynamics (SPH) approach in thermo-fluid application has been starting to gain popularity. Depending on the SPH boundary condition treatment, different methods can be devised to compute the total wall heat transfer rate. In this paper, for the first time, the accuracies of using the popular dummy particle methods, i.e. (a) the Adami Approach (AA) and (b) the higher-order mirror + Moving Least Square (MMLS) method in predicting the total wall heat transfer rate are comprehensively assessed. The modified equation of the 1D wall heat transfer rate is formulated using Taylor's series. For uniform particle layout, MMLS is first-order accurate. Nevertheless, for an irregular particle layout, its order of accuracy drops to ~O(1), the order similar to that of the computationally simpler AA. The AA method is then used to simulate several steady and unsteady natural convection problems involving convex and concave wall geometries. The estimated wall heat transfer rate and the flow results agree considerably well with the available experimental data and benchmark numerical solutions. In general, the current work shows that AA can offer a practical means of estimating wall heat transfer rate at reasonable accuracy for problems involving complex geometry.

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

This paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle.

Improved treatment of wall boundary conditions for a particle method with consistent spatial discretization

In recent years, consistent spatial discretization schemes for meshfree particle methods to numerically simulate incompressible flow have been studied by many researchers. This study is focused on the treatment of solid wall boundary conditions for one of those schemes, namely the least squares MPS (LSMPS) scheme, and proposes a new technique to deal with no-slip and free-slip wall boundary conditions. With the proposed treatment, wall geometries are expressed by surface meshes, i.e. polygons in 3D and line segments in 2D. Thus, complicated geometries can be handled easily. Based on a Taylor series expansion, wall boundary conditions are incorporated into the differential operators acting on fluid particles located in the vicinity of a wall, through a least squares approach. As a consequence, Neumann boundary conditions can be treated quite efficiently. To verify consistency of the proposed discretization scheme, a convergence study was carried out. As numerical examples, Couette flow, plane Poiseuille flow, gravity-driven flow in a 3D square duct, a rigid rotation problem, Taylor–Green vortices and lid-driven cavity flow have been calculated using the proposed boundary treatment, with both no-slip and free-slip conditions applied. As a result, the present method agreed well with the reference solutions, which verified its computational accuracy.

A Numerical Study of Fluid Flow in a Vertical Slot Fishway with the Smoothed Particle Hydrodynamics Method

Fishways have a great ecological importance as they help mitigate the interruptions of fish migration routes. In the present work, the novel DualSPHysics v4.4 solver, based on the smoothed particle hydrodynamics method (SPH), has been applied to perform three-dimensional (3-D) simulations of water flow in a vertical slot fishway (VSF). The model has been successfully calibrated against published field data of flow velocities that were measured with acoustic Doppler velocity probes. A state-of-the-art algorithm for the treatment of open boundary conditions using buffer layers has been applied to accurately reproduce discharges, water elevations, and average velocity profiles (longitudinal and transverse velocities) within the observed pool of the VSF. Results herein indicate that DualSPHysics can be an accurate tool for modeling turbulent subcritical free surface flows similar to those that occur in VSF. A novel relation between the number of fluid particles and the artificial viscosity coefficient has been formulated with a simple logarithmic fit.

Adjoint-based optimization of sound reinforcement including non-uniform flow.

The determination of optimal geometric arrangements and electronic drives of loudspeaker arrays in sound reinforcement applications is an ill-posed inverse problem. This paper introduces an innovative method to determine complex driving functions, also considering complex environmental conditions. As an alternative to common frequency domain methods, the authors present an adjoint-based approach in the time domain: Acoustic sources are optimized in order to generate a given target sound field. Instead of the Helmholtz equation, the full non-linear Euler equations are considered. This enables an easier treatment of non-uniform flow and boundary conditions. As proof of concept, a circular and a linear monopole array are examined. For the latter, the environmental conditions include wind and thermal stratification. For all examples, the method is able to provide appropriate driving functions.

On the implementation and consequences of the oceanic currents feedback in ocean–atmosphere coupled models

The Current FeedBack (CFB) to the atmosphere simply represents the influence of the surface oceanic currents on near-surface wind and surface stress. As the CFB has a significant influence on the oceanic circulation, it is crucial to implement it properly in a coupled ocean–atmosphere model. In this study, we first detail the modifications to be implemented into atmospheric models to account for the CFB. In the computation of air–sea fluxes, the relative winds, i.e., the difference between the near-surface winds and the surface oceanic currents, instead of absolute winds have to be used (Modification M1). However, because of the implicit treatment of the bottom boundary condition in most atmospheric models, the use of relative winds also involves a modification of the tridiagonal problem associated with the discretization of the vertical turbulent viscosity (Modification M2). Secondly, we show both analytically and using global coupled simulations that omitting M2 leads to a large underestimation of the surface stress curl response to the CFB and, subsequently, of the coupling coefficient between mesoscale surface stress curl and surface current vorticity. As a consequence, the dampening of the mesoscale activity induced by the CFB is strongly reduced (by a minimum a factor of 2 or more). The practical implementation of the CFB must be done carefully in the atmospheric component of a coupled model in order to avoid a large underestimation of the CFB effect on the oceanic circulation.