Local Stencils Research Articles

4-dimensional local radial basis function interpolation of large, uniformly spaced datasets

Background and ObjectiveLarge, uniformly spaced, complex and time varying datasets derived from high resolution medical image velocimetry can provide a wealth of information regarding small-scale transient physiological flow phenomena and pulsation of anatomical boundaries. However, there remains a need for interpolation techniques to effectively reconstruct a fully 4-dimensional functional relationship from this data. This paper presents a preliminary evaluation of a 4-dimensional local radial basis function (RBF) algorithm as a means of addressing this problem for laminar flows. MethodsA 4D interpolation algorithm is proposed based on a Local Hermitian Interpolation (LHI) using a combination of multi-quadric RBF with a partition of unity scheme. The domain is divided into uniform sub-systems with size restricted to immediately neighbouring points. The validity of the algorithm is first established on a known 4D analytical dataset and a CFD based laminar flow phantom. Application is then demonstrated through characterisation of a large 4D laminar flow dataset obtained from magnetic resonance imaging (MRI) measurements of cerebrospinal fluid velocities in the brain. ResultsPerformance of the algorithm is compared to that of a quad-linear interpolation, demonstrating favourable improvement in accuracy. The technique is shown to be robust, computationally efficient and capable of refined interpolation in Euclidean space and time. Application to MR velocimetry data is shown to produce promising results for the 4D reconstruction of the transient flow field and movement of the fluid boundaries at spatial and temporal locations intermediate to the original data. ConclusionThis study has demonstrated feasibility of an accurate, stable and efficient 4-dimensional local RBF interpolation method for large, transient laminar flow velocimetry datasets. The proposed approach does not suffer from ill-conditioning or high computational cost due to domain decomposition into local stencils where the RBF is only ever applied to a limited number of points. This work offers a potential tool to assist medical diagnoses and drug delivery through better understanding of physiological flow fields such as cerebrospinal fluid. Further work will evaluate the technique on a wider range of flow fields and against CFD simulation.

A higher-order accurate VOF interface curvature computation scheme for 3D non-orthogonal structured meshes

This paper proposes a higher-order accurate curvature computation method for volume-of-fluid (VOF) interfaces on 3D non-orthogonal structured meshes. Analogous to the height-function (HF) method, a novel approach is introduced to identify columns of control volumes that straddle the interface. This is followed by a volume conserving piecewise-linear interface construction (PLIC). In the interest of efficiency and generality, the latter employs a novel sweep-plane algorithm with bracketing on convex-decomposed control volumes in each column. The emphasis is on the generalisation of a PLIC-based least squares polynomial surface fitting procedure owing to its invariance with respect to local coordinate rotations in 3D. The PLIC representation of the interface is then smoothed by iteratively refining the PLIC facet normals. The interface curvature is finally computed analytically by fitting either second- or fourth-order polynomial surfaces to local stencils of PLIC facets in a least squares manner. Formal second- and fourth-order accuracy of interface curvature is demonstrated by comparing numerical results with a variety of analytical interface definitions.

DCGrid

We introduce Dynamic Constrained Grid (DCGrid), a hierarchical and adaptive grid structure for fluid simulation combined with a scheme for effectively managing the grid adaptations. DCGrid is designed to be implemented on the GPU and used in high-performance simulations. Specifically, it allows us to efficiently vary and adjust the grid resolution across the spatial domain and to rapidly evaluate local stencils and individual cells in a GPU implementation. A special feature of DCGrid is that the control of the grid adaption is modeled as an optimization under a constraint on the maximum available memory, which addresses the memory limitations in GPU-based simulation. To further advance the use of DCGrid in high-performance simulations, we complement DCGrid with an efficient scheme for approximating collisions between fluids and static solids on cells with different resolutions. We demonstrate the effectiveness of DCGrid for smoke flows and complex cloud simulations in which terrain-atmosphere interaction requires working with cells of varying resolution and rapidly changing conditions. Finally, we compare the performance of DCGrid to that of alternative adaptive grid structures.

A higher-order accurate surface tension modelling volume-of-fluid scheme for 2D curvilinear meshes

This paper proposes a higher-order accurate curvature calculation method for volume-of-fluid (VOF) based surface tension modelling on 2D curvilinear meshes. The scheme extends the interface reconstruction component of the height function (HF) method to curvilinear meshes in 2D. Linear reconstructions are computed in each column overlapping the interface, which results in a piecewise-linear (PLIC) representation of the free-surface. This is done in a volume conservative manner by employing a novel sweep-line algorithm. The interface curvature is then computed analytically by fitting either second- or fourth-order polynomials to local stencils of PLIC facets. Formal second- and fourth-order accuracy of interface curvature is demonstrated by comparing numerical results with a variety of analytical interface definitions. The curvature algorithm is implemented into a balanced-force continuum-surface-force surface tension scheme for variable-density flows. Second- and fourth-order accuracy are again demonstrated for the Laplace pressure jump in the simulation of a stationary bubble with a high liquid-gas density ratio. Lastly, second-order accuracy is demonstrated in computing the frequency of an inviscid oscillating droplet in zero gravity.

Multiscale Polynomial-Based High-Order Central High Resolution Schemes

Polynomial-based high order central high resolution schemes with semi-discrete forms are integrated with multiresolution-based adapted cells/grids. To preserve the positivity condition on non-uniform cells/grids, the corresponding formulations are studied, redesigned or developed. Two general approaches can be used for polynomial-based reconstructions: (a) direct interpolation by a polynomial, (b) proper combination of different polynomials to construct a new polynomial with desired features. Based on these approaches, three polynomial-based reconstructions are considered: (i) parabolic polynomials interpolating average solutions of three successive cells; (ii) piece-wise parabolic methods (PPMs) obtained with two different local stencils; (iii) central-WENO schemes [based on the results of approach (i)]. For the first approach, the corresponding features, stability conditions, formulations and nonlinear limiters are studied and updated. For the second approach, for more localized stencils, new independent variables (e.g., first and second spatial derivatives) are introduced by adding new conservation laws. Two PPM-based central schemes are formulated and a new limiter and a new updating procedure are introduced. For the third approach, the average-interpolating parabolic polynomial [in approach (i)] is used in the framework of the central-WENO formulation. Third and fourth order formulations are provided on non-uniform grids/cells. Finally some numerical examples are presented to verify the results and to assess effectiveness and robustness of the three approaches.

A local meshless method based on the finite collocation and local integral equations method for delay PDEs

A local meshless method based on the finite collocation, local radial basis function (RBF) and MLPG method is proposed for solving the 2D Delay partial differential equations. In fact, instead of collocation of governing equations in the finite collocation method, we propose the use of local weak form of governing equations on the local stencils. The method is used for the numerical solutions of the 2D delay partial differential equations (PDEs). Two delay and one nonlinear multidelay parabolic PDEs are solved as test problems and the comparisons of the results reveal the effectiveness of the method.

Stream Line–Based Pattern Search in Flows

AbstractWe propose a method that allows users to define flow features in form of patterns represented as sparse sets of stream line segments. Our approach finds similar occurrences in the same or other time steps. Related approaches define patterns using dense, local stencils or support only single segments. Our patterns are defined sparsely and can have a significant extent, i.e., they are integration‐based and not local. This allows for a greater flexibility in defining features of interest. Similarity is measured using intrinsic curve properties only, which enables invariance to location, orientation, and scale. Our method starts with splitting stream lines using globally consistent segmentation criteria. It strives to maintain the visually apparent features of the flow as a collection of stream line segments. Most importantly, it provides similar segmentations for similar flow structures. For user‐defined patterns of curve segments, our algorithm finds similar ones that are invariant to similarity transformations. We showcase the utility of our method using different 2D and 3D flow fields.

A high‐resolution local RBF collocation method for steady‐state poroelasticity and hydromechanical damage analysis

SummaryIn this work, we describe a meshless numerical method based on local collocation with RBFs for the solution of the poroelasticity equation. The RBF finite collocation approach forms a series of overlapping nodal stencils, over which an RBF collocation is performed. These local collocation systems enforce the governing PDE operator throughout their interior, with the intersystem communication occurring via the collocation of field variables at the stencil periphery. The method does not rely on a generalised finite differencing approach, whereby the governing partial differential operator is reconstructed at the global level to drive the solution of the PDE. Instead, the PDE governing and boundary operators are enforced directly within the local RBF collocation systems, and the sparse global assembly is formed by reconstructing the value of the field variables at the centrepoint of the local stencils. In this way, the solution of the PDE is driven entirely by the local RBF collocation, and the method more closely resembles the approach of the full‐domain RBF collocation method. By formulating the problem in this fashion, high rates of convergence may be attained without the computational cost and numerical ill‐conditioning issues that are associated with the full‐domain RBF collocation approach.An analytical solution is formulated for a 2D poroelastic fluid injection scenario and is used to verify the proposed implementation of the method. Highly accurate solutions are produced, and convergence rates in excess of sixth order are observed for each field variable (i.e. pressure and displacement) and field‐variable derivative (i.e. pressure gradients and stresses). The stress and displacement fields resulting from the solution of the poroelasticity equation are then used to describe the formation and propagation of microfractures and microfissures, which may form in the presence of large shear strain, in terms of a continuous damage variable which modifies the mechanical and hydraulic properties of the porous medium. The formation of such hydromechanical damage, and the resulting increase in hydraulic conductivity, is investigated for a pressurised injection into sandstone. Copyright © 2014 John Wiley & Sons, Ltd.

Efficient fast marching with Finsler metrics

We study the discretization of the escape time problem: find the length of the shortest path joining an arbitrary point \(z\) of a domain \(\Omega \), to the boundary \(\partial \Omega \). Path length is measured locally via a Finsler metric, potentially asymmetric and strongly anisotropic. This optimal control problem can be reformulated as a static Hamilton–Jacobi partial differential equation, or as a front propagation model. It has numerous applications, ranging from motion planning to image segmentation. We introduce a new algorithm, fast marching using anisotropic stencil refinement (FM-ASR), which addresses this problem on a two dimensional domain discretized on a cartesian grid. The local stencils used in our discretization are produced by arithmetic means, like in the FM-LBR (Mirebeau in Anisotropic fast Marching on Cartesian grids, using lattice basis reduction, preprint 2012), a method previously introduced by the author in the special case of Riemannian metrics. The complexity of the FM-ASR, in an average sense over all grid orientations, only depends (poly-)logarithmically on the anisotropy ratio of the metric, while most alternative approaches have a polynomial dependence. Numerical experiments show, in several occasions, that the accuracy/complexity compromise is improved by an order of magnitude or more.

Compact local IRBF and domain decomposition method for solving PDEs using a distributed termination detection based parallel algorithm

Compact Local Integrated Radial Basis Function (CLIRBF) methods based on Cartesian grids can be effective numerical methods for solving partial differential equations (PDEs) for fluid flow problems. The combination of the domain decomposition method and function approximation using CLIRBF methods yields an effective coarse-grained parallel processing approach. This approach has enabled not only each sub-domain in the original analysis domain to be discretised by a separate CLIRBF network but also compact local stencils to be independently treated. The present algorithm, namely parallel CLIRBF, achieves higher throughput in solving large scale problems by, firstly, parallel processing of sub-regions which constitute the original domain and, secondly, accelerating the convergence rate within each sub-region using groups of CLIRBF stencils in which function approximations are carried out by parallel processes. The procedure is illustrated with several numerical examples of PDEs and lid-driven cavity problem using Message Passing Interface supported by MATLAB.

A gradient-augmented level set method with an optimally local, coherent advection scheme

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.