Interpolation Stencil Research Articles

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in \(\mathbb {R}^2 \) or \(\mathbb {R}^3 \) , such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

Large-eddy simulation of wall-bounded incompressible turbulent flows based on multi-moment finite volume formulation

The multi-moment based numerical model is proposed for performing large-eddy simulation (LES) of wall-bounded incompressible turbulent flows on the unstructured grids. In this model, velocity is defined as volume integral average (VIA) and point values (PVs) which are updated as prognostic variables simultaneously in time. Using VIA and PV in each cell, this method constructs higher-order polynomial on a compact stencil and more importantly, provides a new approach to design subgrid-scale (SGS) models for LES. Various SGS models are developed in this framework where the eddy-viscosity is computed separately at both VIA and PVs using the least-square method on different interpolation stencils. Compared with conventional finite volume schemes, the proposed method effectively reduces numerical dissipation and largely suppresses the sensitivity of numerical solution on SGS models and resolves more elaborated flow structures on the same mesh resolution. It also reduces solution dependence on the shape and quality of grid elements, as demonstrated by comparing numerical results on the different types of grids. The proposed method can attain high-fidelity simulation without loss of numerical efficiency and robustness, which is highly appealing for complex turbulent problems with high Reynolds numbers and complicated geometries.

Implicit interpolation method for immersed boundary methods

Immersed boundary (IB) methods have been successfully implemented for different applications. This paper focuses on the immersed boundary implementation for two different governing equations, namely the diffusion equation and Euler equations, using a bi-linear interpolation for the implementation of the boundary condition. The concept of implicit interpolation is introduced which eradicates the problems faced with the explicit interpolation in which it is required to move away from the boundary in the fluid domain in order to complete the interpolation stencil.

Grid-Characteristic Numerical Method on an Irregular Grid with Extending the Interpolation Stencil

A grid-characteristic numerical method for solving a multidimensional transport equation on an unstructured grid with an order higher than one is proposed; this method does not use auxiliary points on edges and faces. The avoidance of auxiliary points on edges and faces simplifies the topology of the computational grid during its motion, which is important for solving dynamic problems of mechanics of deformable solids. To increase the approximation order, an analog of the grid stencil extension implemented for an unstructured grid is used. Results of testing the proposed numerical scheme for continuously differentiable, continuous, discontinuous solutions are presented.

Grid-Characteristic Numerical Method on an Irregular Grid with Extending the Interpolation Stencil

Grid-Characteristic Numerical Method on an Irregular Grid with Extending the Interpolation Stencil

Comparing two IBM implementations for the simulation of uniform packed beds

Nowadays, the design of fixed packed bed reactors still relies on empirical correlations, which, especially for small tube to particle diameter ratios, are mostly too inaccurate because of the presence of wall effects. Therefore, the simulation of fixed packed bed reactors plays an important role to predict and control the flow and process parameters in such, nowadays and in the future. Because of its straightforward applicability to non-uniform packings with particles of arbitrary shapes, the immersed boundary method (IBM) has advantages over other numerical methods and is used more and more frequently. This paper compares two approaches of IBMs for the simulation of fixed bed reactors with spherical shaped particles. The classic, smooth approach is compared to the straightforward to implement blocked-off method for velocity fields above the fixed bed for particle Reynolds numbers of 300 and 500. Results from experimental inline PIV-measurements of the reactor to be simulated serve as a basis for comparison. Very good agreement with the experiment is found for both simulation methodologies with higher resolutions, considering the more stable flow at a particle Reynolds number of 300. Differences in the different IBM approaches occurred for the more unsteady flow at a particle Reynolds number of 500. Compared to the blocked-off method, the smooth IBM reflects the formation of additional jets and recirculation zones better right above the bed, though increasing the fluid mesh resolution improves the accuracy of the blocked-off method. Overall, a more diffusive behaviour is found for the blocked-off simulations due to the stairstep representation, which is avoided by using interpolation stencils as in the smooth IBM. With higher mesh refinement in the blocked-off IBM this effect can be reduced, but this also increases the computational effort.

Efficient Parallel Dynamic Overset Unstructured Grid for Flow Simulation with Moving Bodies

The overset grid method for simulation of unsteady flow with moving bodies faces several issues, including low assembly efficiency for a large number of bodies, difficult parallel implementation, and the requirement of manual intervention. To address these issues, in this paper, we develop an efficient, automatic, and robust parallel dynamic overset-unstructured-grid assembly method. It is achieved by parallelizing its two main steps: hole cutting, and identification of interpolation stencils. The hole cutting is simply implemented in parallel by identifying the active zone of each grid with a criterion of the global minimum wall distance. The global minimum wall distance is efficiently calculated on each process with the oriented-bounding-box-based -dimensional (-D) trees of the wall surface grids. For identification of the interpolation stencil, we use the results of the first step, optimally define the interpolation boundary nodes, and determine their candidate donor partitions in each process. This efficiently minimizes the number of query nodes and their donor cell candidates. The donor cell search is efficiently performed in parallel in each candidate donor partition by using the oriented-bounding-box-based -D tree of the field volume grid. Several cases are adopted to test the efficiency and capability of the proposed parallel dynamic overset-unstructured-grid assembly approach.

A high-efficiency discretized immersed boundary method for moving boundaries in incompressible flows

The Immersed Boundary Method (IBM) has an advantage in simulating fluid–structure interaction, owning to its simplicity, intuitiveness, and ease of handling complex object boundaries. The interpolation function plays a vital role in IBM and it is usually computationally intensive. For moving or deforming solids, the interpolation weights of all the immersed boundary points ought to be updated every time step, which takes quite a lot CPU time. Since the interpolation procedure within all uniform structured grids is highly repetitive and very similar, we propose a simple and generalized Discretized Immersed Boundary Method (DIBM), which significantly improves efficiency by discretizing the interpolation functions onto subgrid points within each control volume and reusing a predefined universal interpolation stencil. The accuracy and performance of DIBM are analyzed using both theoretical estimation and simulation tests. The results show speedup ratios of 30–40 or even higher using DIBM when compared with conventional IBM for typical moving boundary simulations like particle-laden flows, while the error is estimated to be under 1% and can be further decreased by using finer subgrid stencils. By balancing the performance and accuracy demands, DIBM provides an efficient alternative framework for handling moving boundaries in incompressible viscous flows.

A face-based immersed boundary method for compressible flows using a uniform interpolation stencil

In this study, an immersed boundary method developed for compressible viscous flows (Ramakrishnan, R., Girdhar, A., & Ghosh, S. (2016). Immersed Boundary Methods for Compressible Laminar Flows) is modified to improve their stability and robustness. The embedded object is represented as a set of line segments in two dimensions with their outward unit normal vectors specified. A forcing method that leverages the finite volume approach is used, wherein the solution at the cell interfaces that lie near the boundaries of the embedded solid is reconstructed to implicitly satisfy boundary conditions at the immersed surface. The proposed immersed boundary method is validated for transonic inviscid flow past a bump in a channel, supersonic flow past a circular cylinder, transonic viscous flow past a NACA0012 airfoil, and supersonic viscous flow past a circular cylinder. The results are compared with simulations from the literature using contours of flow properties, surface pressure, or Mach number plots and show good agreement.

Разработка явных и консервативных схем для решеточных уравнений Больцмана с адаптивным переносом

The Lattice Boltzmann Method (LBM) has several limitations for velocity and temperature. One can consider distribution function in moving frame to overcome these limitations as in PonD. In PonD, values of distribution functions are streamed from off-lattice points, so value estimation is needed. It leads to the implicit and non-conservative numerical scheme. Earlier, for the one-dimensional case, the approach of moments prediction was found, which leads to an explicit and conservative numerical scheme. We apply this approach to the two-dimensional and three-dimensional cases in this work. Requirements to interpolation stencil, quadrature, and Hermite polynomial expansion which guarantee moment matching, conservation, and exact calculation, were studied. The resulting schemes were implemented and tested on several tasks.

Nonlinear convergence in contact mechanics: Immersed boundary finite volume

We present an immersed boundary finite volume (IBM) method for simulating quasistatic contact mechanics of linearly elastic domains at small strains. In IBM, all external boundaries and internal contacts of an object are represented by embedded surfaces inside a Cartesian mesh, which need not conform to the grid lines. The contact constraints consist of the non-penetrability condition and Coulomb’s friction law, which are discretized using special interpolation stencils and enforced via penalty parameters. The resulting nonlinear system depends on displacement unknowns only. To solve it, we use the Newton method but find that it diverges frequently. To understand the divergence pattern, we analyze a simplified 2-cell problem and show that the global convergence of Newton cannot be ensured for any choice of penalty parameters. We thus propose a modified Newton solver, which guarantees convergence for the 2-cell problem and is numerically verified to converge for all the challenging simulations considered herein. While both 1st- and 2nd-order variants of IBM, in displacement unknowns, are proposed, the modified Newton solver applies only to the 1st-order variant.

High order ghost-cell immersed boundary method for generalized boundary conditions

Flow and reactive transport problems in engineering, medical and environmental applications often involve complex geometries. Grid based methods (e.g. finite volume, finite element, etc.) are a vital tool for studying such problems. Cartesian grids are one of the most attractive options as they possess simple discretization stencils and are usually straightforward to generate at roughly no computational cost. The Immersed Boundary Method, a Cartesian based methodology, maintains most of the useful features of structured grids, while it exhibits a great resilience in dealing with complex geometries. These features make it increasingly more attractive to model transport in evolving porous media as the cost of grid generation reduces greatly. Yet, stability issues due to the geometry of the interpolation stencil combined with limited studies on the implementation of Neumann (constant flux) and linear Robin (e.g. reaction) boundary conditions have significantly limited its applicability to transport in complex topologies. We develop a high-order compact Cartesian model based on ghost cell immersed boundary method for incompressible flow and scalar transport subject to different boundary conditions. The accuracy test shows at least second order of accuracy in L1,L2 and L∞ norms of error. The proposed method is capable of accurately capturing the transport physics near the boundaries for Dirichlet, Neumann and Robin boundary conditions. We tested the method for several transport and flow scenarios, including heat transfer close to an immersed object and mass transport over reactive surfaces.

An immersed boundary method for flows with dense particle suspensions

The immersed boundary method (IBM) is a well-suited tool for the direct numerical simulation of flows with dense particle suspensions, which feature strong and complex flow-particle and particle–particle interactions. With the aim to model such flows, this paper proposes to extend the Regularised Discrete Momentum Forcing (RDMF-) IBM of Abdol Azis et al. (2018) to immersed boundaries (IBs) whose motion is coupled to the flow. A modification to the direct-forcing formulation, resulting in an offset of the imposed no-slip boundary from a moving immersed boundary, is proposed. This offset is exploited in an iterative strong flow-particle coupling scheme, where the boundary forces are applied at the location of the IB known from the previous time level, while correctly enforcing the no-slip condition at the new location of the IB. This avoids having to reconstruct the interpolation and spreading stencils for every linear iteration of the flow-particle coupling, therefore reducing the computational load of the method. A technique to ensure the stable modelling of flows involving light particles is also presented and is shown to stably cope with a wide range of particle–fluid density ratios. The framework is validated by comparing the results of the simulation of a single sedimenting sphere with published experimental results. Simulations of the fluidisation of heavy rigid particles, as well as of the ascent of closely packed light particles, are also presented. Use of the radius retraction procedure, to compensate for the diffuse representation of the smooth-interface IBM, is demonstrated to consistently yield significant improvements for moving IB cases with low to intermediate Reynolds numbers. The present framework shows good stability properties for a wide range of flow regimes with concentrated particle suspensions.

Hybrid Optimized Low-Dissipation and Adaptive MUSCL Reconstruction Technique for Hyperbolic Conservation Laws

A new hybrid optimized low-dissipation and adaptive MUSCL scheme is present for finite volume method. The proposed scheme, based on an optimized linear scheme with monotonicity preserving limitation and an adaptive MUSCL scheme, emphasizes on the resolution rather than the formal order. This technique is applied to cell interface reconstruction and bears similarity in form to the widely used MUSCL scheme except wider interpolation stencil. Although the scheme is not high order of accuracy because of adaptive MUSCL scheme acting as nonlinear part, the low-dissipation feature from the optimized linear part makes it very accurate and robust in many practical applications, where much richer flow structures can be obtained. A number of test cases are solved to validate the high resolution of the present scheme.

Extension of the unified interpolation stencil for immersed boundary method for moving boundary problems

Extension of the unified interpolation stencil for immersed boundary method for moving boundary problems

A Second Order Ghost Fluid Method for an Interface Problem of the Poisson Equation

AbstractA second order Ghost Fluid method is proposed for the treatment of interface problems of elliptic equations with discontinuous coefficients. By appropriate use of auxiliary virtual points, physical jump conditions are enforced at the interface. The signed distance function is used for the implicit description of irregular domain. With the additional unknowns, high order approximation considering the discontinuity can be built. To avoid the ill-conditioned matrix, the interpolation stencils are selected adaptively to balance the accuracy and the numerical stability. Additional equations containing the jump restrictions are assembled with the original discretized algebraic equations to form a new sparse linear system. Several Krylov iterative solvers are tested for the newly derived linear system. The results of a series of 1-D, 2-D tests show that the proposed method possesses second order accuracy in L∞ norm. Besides, the method can be extended to the 3-D problems straightforwardly. Numerical results reveal the present method is highly efficient and robust in dealing with the interface problems of elliptic equations.

A unified interpolation stencil for ghost-cell immersed boundary method for flow around complex geometries

A unified interpolation stencil for ghost-cell immersed boundary method for flow around complex geometries

A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods

Most implementations of meshless BEMs use a circular integration contours (spherical in 3D) embedded into a local interpolation stencil with the so-called Companion Solution (CS) as a kernel, in order to eliminate the contribution of the single layer potential. However, the Dirichlet Green's Function (DGF) is the unique Fundamental Solution that is identically zero at any given close surface and therefore eliminates the single layer potential. One of the main objectives of this work is to show that the CS is nothing else than the DGF for a circle collocated at its origin. The use of the DGF allows the collocation at more than one point, permitting the implementation of a P-adaptive scheme in order to improve the accuracy of the solution without increasing the number of subregions. In our numerical simulations, the boundary conditions are imposed at the interpolation stencils in contact with the problem boundary instead of at the corresponding integration surfaces, permitting always the use of circular integration contours, even in regions near or in contact with the problem domain where the densities of the integrals are reconstructed from the interpolation formulae that already included the problem boundary conditions.

Efficiency benchmarking of seventh-order tri-diagonal weighted compact nonlinear scheme on curvilinear mesh

ABSTRACTEfficiency improvements of high-order weighted compact nonlinear scheme (WCNS) are verified using a series of benchmark cases, proposed at the International Workshop on High-Order CFD Methods. A seventh-order tri-diagonal compact one of WCNSs (WCNS-E8T7), constructed in recent years, is investigated as a basic scheme, and compared to a typical fifth-order explicit WCNS (WCNS-E6E5) and a traditional second-order TVD scheme MUSCL. Among these tests, a symmetrical conservative metric method (SCMM) is adopted to ensure the accuracy and robustness of WCNSs when solving cases in the curvilinear coordinates. The computational efficiency of schemes is evaluated based on a non-dimensional cost in achieving the same level of accuracy. Related results show that WCNS-E8T7 has a better performance than WCNS-E6E5 with the same interpolation stencils. Moreover, the opinion that high-order methods can obtain higher computational efficiency than second-order methods is demonstrated on the cases ranging from academic problems to real-life computations.

Simplex-stochastic collocation method with improved scalability

The Simplex-Stochastic Collocation (SSC) method is a robust tool used to propagate uncertain input distributions through a computer code. However, it becomes prohibitively expensive for problems with dimensions higher than 5. The main purpose of this paper is to identify bottlenecks, and to improve upon this bad scalability. In order to do so, we propose an alternative interpolation stencil technique based upon the Set-Covering problem, and we integrate the SSC method in the High-Dimensional Model-Reduction framework. In addition, we address the issue of ill-conditioned sample matrices, and we present an analytical map to facilitate uniformly-distributed simplex sampling.