Partial Differential Equation Boundary Value Research Articles

Mesh-Clustered Gaussian Process Emulator for Partial Differential Equation Boundary Value Problems

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this article, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

Deep Learning Methods for Reynolds-Averaged Navier–Stokes Simulations of Airfoil Flows

This study investigates the accuracy of deep learning models for the inference of Reynolds-averaged Navier–Stokes (RANS) solutions. This study focuses on a modernized U-net architecture and evaluates a large number of trained neural networks with respect to their accuracy for the calculation of pressure and velocity distributions. In particular, it is illustrated how training data size and the number of weights influence the accuracy of the solutions. With the best models, this study arrives at a mean relative pressure and velocity error of less than 3% across a range of previously unseen airfoil shapes. In addition all source code is publicly available in order to ensure reproducibility and to provide a starting point for researchers interested in deep learning methods for physics problems. Although this work focuses on RANS solutions, the neural network architecture and learning setup are very generic, and applicable to a wide range of partial differential equation boundary value problems on Cartesian grids.

Analysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical method

A hybrid numerical method is developed for three-dimensional (3D) heat conduction in functionally graded materials (FGMs) by integrating the advantages of the generalized finite difference method (GFDM) and Krylov deferred correction (KDC) technique. The temporal direction of the problems is first discretized by applying the KDC approach for high-accuracy results, which yields a partial differential equation (PDE) boundary value problem at each time step. The corresponding PDE boundary value problem is then solved by introducing the meshless GFDM that has no requirement of time-consuming meshing generation and numerical integration for 3D problems with complex geometries. Numerical experiments with four types of material gradations are provided to verify the developed combination scheme, and numerical results demonstrate that the method has a great potential for 3D transient heat conduction in FGMs especially for those in a long-time simulation.

Computational unsteady flow analysis for third‐grade fluid from an isothermal vertical cylinder through a Darcian porous medium

AbstractThe present study describes a mathematical model for free‐convective laminar incompressible boundary layer flow of a third‐grade fluid of the Reiner‐Rivlin differential type, external to an evenly heated semi‐infinite vertical cylinder through a two‐dimensional porous medium. Assuming a homogenous‐isotropic porous medium, simulation of bulk drag effects at low Reynolds number is conducted with the Darcy model. The resulting partial differential equation boundary value problem is normalized using suitable transformation variables. The highly nonlinear time‐dependent coupled conservation equations along with boundary conditions are resolved computationally with an optimized Crank‐Nicolson finite difference code. Validation with previous studies is included. The heat transport and skin‐friction coefficients are computed for different values of emerging nondimensional parameters. Furthermore, steady‐state and transient fluid flow variables are shown graphically. An enhanced fluid velocity is observed for increased Darcy number and the reverse trend is computed for higher values of third‐grade viscoelastic parameter. Also, the rate of heat transfer is observed to increase with greater Darcy number and a reduction in third‐grade viscoelastic parameter. A key observation which is drawn from the present study is that for third‐grade fluid the flow variables deviate significantly from a hot cylindrical wall as compared with a Newtonian fluid. The study is relevant to thermal polymer coating applications in aerospace materials processing.

Efficient Distribution Estimation and Uncertainty Quantification for Elliptic Problems on Domains with Stochastic Boundaries

We study the problem of uncertainty quantification for the numerical solution of elliptic partial differential equation boundary value problems posed on domains with stochastically varying boundari...

On the breakdown of Rayleigh’s criterion for curved shear flows: a destabilization mechanism for a class of inviscidly stable flows

AbstractThe stability of a high-Reynolds-number flow over a curved surface is considered. Attention is focused on spanwise-periodic vortices of wavelength comparable with the boundary layer thickness. The wall curvature and the Görtler number for the flow are assumed large, so that stable or unstable vortices of wavelength comparable with the boundary layer thickness are dominated by inviscid effects. The growth or decay rate determined by using the viscous correction to the inviscid prediction is found to give a good approximation to the exact value for a range of wavenumbers. For the stable configuration, it is shown that the flow can be destabilized by arbitrarily small wall waviness, with the critical configuration involving only geometric properties of the flow. This destabilization mechanism has consequences for a number of flows of aerodynamic interest, particularly those where surface waviness under loading occurs. Control strategies based on how the curvature is distributed are discussed, and it is demonstrated how small regions of varying curvature can be used to remove the exponentially growing modes. The development of the parametric instabilities is continued into the strongly nonlinear regime. The nonlinear evolution is found to be governed by a generalized form of the cubic amplitude equation, generally referred to as the Stuart–Landau equation. The novel feature of the new equation is that the nonlinearity is not in the form of a power of the amplitude but is fixed by the solution of an associated nonlinear partial differential equation boundary value problem. Numerical solutions of the nonlinear partial differential system that fix the disturbance amplitude are given. The mechanism of destabilization described is shown to be directly relevant to a number of flows that are inviscidly stable in the presence of body forces. Thus it is shown how Rayleigh–Bénard convection or Taylor vortices can, at high Rayleigh or Taylor numbers, be destabilized by asymptotically small modulation in time or space.

Dynamic Moisture Absorption Behavior of Polyester-Cotton Fabric and Mathematical Model

We investigate the dynamic moisture absorption behavior of polyester—cotton fabrics of different warp and weft densities, survey the dry-basis moisture contents of these various fabrics under variant temperature and relative humidity, and acquire a dynamic moisture absorption curve and balanced moisture content of these fabrics. The results show that the fabric moisture absorption velocity is in reverse relation with its warp and weft densities. By applying the Fick’s Law for diffusion in porous media, a mathematical model for the fabric’s dynamic moisture absorption is established, which is represented in the form of a non-linear partial differential equation boundary value problem; in order to simplify the calculation of the solution, a model parameter a is put forward, defined as the “structural factor”, relating to the fabric structure. After the model is solved under six-point format finite difference method, the distribution of the dry-basis moisture content in the diffusion direction at different times can be obtained. Based on the experimental data, the golden section method is utilized to estimate the model parameter a, and the linear relation between the parameter a and the product of warp and weft densities j is obtained by mathematical software “Mathematica”. The simulation value of dry-basis moisture content from the model matches the experimental observations.

A Curvilinear Tool-Path Method for Pocket Machining

A novel curvilinear tool-path generation method is described for planar milling of pockets. The method uses the solution of an elliptic partial differential equation boundary value problem defined on a pocket region. This mathematical function helps morph a smooth low-curvature spiral path in a pocket interior to one that conforms to the pocket boundary. This morphing leads to substantial reductions of tool wear in cutting hard metals and of machining time in cutting all metals, as experiments described here show. A variable feed-rate optimization procedure is also described. This procedure incorporates path, tool-engagement, and machine constraints and can be applied to maximize machine performance for any tool path.

Three parallel programming paradigms: comparisons on an archetypal PDE computation

Three paradigms for distributed-memory parallel computation that free application programmer from details of message passing are compared for an archetypal structured scientific computation -- a nonlinear, structured-grid partial differential equation boundary value problem -- using same algorithm on same hardware. All of paradigms -- parallel languages represented by Portland Group's HPF, (semi-)automated serial-to-parallel source-to-source translation represented by CAP-Tools from University of Greenwich, and parallel libraries represented by Argonne's PETSc -- are found to be easy to use for this problem class, and all are reasonably effective in exploiting concurrency after a short learning curve. The level of involvement required by application programmer under any paradigm includes specification of data partitioning, corresponding to a geometrically simple decomposition of domain of PDE. Programming in SPMD style for PETSc library requires writing only routines that discretize PDE and its Jacobian, managing subdomain-to-processor mappings (affine global-to-local index mappings), and interfacing to library solver routines. Programming for HPF requires a complete sequential implementation of same algorithm as a starting point, introduction of concurrency through subdomain blocking (a task similar to index mapping), and modest experimentation with rewriting loops to elucidate to compiler latent concurrency. Programming with CAPTools involves feeding same sequential implementation to CAPTools interactive parallelization system, and guiding source-to-source code transformation by responding to various queries about quantities knowable only at runtime. Results representative of the state of practice for a scaled sequence of structured grid problems are given on three of most important contemporary high-performance platforms: IBM SP, SGI Origin 2000, and CRAYY T3E.

A comparison of PETSc library and HPF implementations of an archetypal PDS computation

Two paradigms for distributed-memory parallel computation that free the application programmer from the details of message passing are compared for an archetypal structured scientific computation—a nonlinear, structured-grid partial differential equation boundary value problem—using the same algorithm on the same hardware. Both paradigms, parallel libraries represented by Argonne's PETSc, and parallel languages represented by the Portland Group's HPF, are found to be easy to use for this problem class, and both are reasonably effective in exploiting concurrency after a short learning curve. The level of involvement required by the application programmer under either paradigm includes specification of the data partitioning (corresponding to a geometrically simple decomposition of the domain of the PDE). Programming in SPMD style for the PETSc library requires writing the routines that discretize the PDE and its Jacobian, managing subdomain-to-processor mappings (affine global-to-local index mappings), and interfacing to library solver routines. Programming for HPF requires a complete sequential implementation of the same algorithm, introduction of concurrency through subdomain blocking (an effort similar to the index mapping), and modest experimentation with rewriting loops to elucidate to the compiler the latent concurrency. Correctness and scalability are cross-validated on up to 32 nodes of an IBM SP2.

Theory and computation of periodic solutions of autonomous partial differential equation boundary value problems, with application to the driven cavity problem

In ordinary differential equations, one distinguishes two cases: autonomous and nonautonomous. Roughly speaking, the theory of the latter is built upon the theory of the former. The same distinction should be applied to partial differential equations, where much less is known. Here I will focus on the question of the generation of periodic solutions for autonomous partial differential equation boundary value problems. Specifically, I consider the incompressible Navier-Stokes equations, and the important driven cavity problem. For simplicity, attention is restricted to two bifurcation parameters, the Reynolds number and the Aspect ratio. Only Dirichlet velocity boundary conditions are considered. Both the known theory and known computational results for the driven cavity are surveyed. The importance of computationally adhering to the div u = 0 condition to accurately simulate unsteady flows which will be qualitatively correct for the incompressible Navier-Stokes equations is stressed. The dependence of sustained periodicity upon the existence of highly localized vortex shedding sequences somewhere along the boundary is pointed out. A new analysis of the pressure boundary condition, based upon a general regularity principle, is given. A conjectured Hopf bifurcation criticality curve is explained.