Velocity interpolation and streamline tracing on irregular geometries

Klausen, Runhild Aae; Rasmussen, A.F.; Stephansen, Annette F.

doi:10.1007/s10596-011-9256-0

Cited by 45 publications

(28 citation statements)

References 24 publications

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“…Muscle fascicle tracts were traced in the flux field by applying a streamline tracing method for divergence free flow fields as described by Klausen et al [32]. In this approach, barycentric coordinates are used for an element-wise tracing which is controlled by the given flux values at the faces of each element.…”

Section: Methodsmentioning

confidence: 99%

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Skeletal Muscle Fascicle Arrangements Can Be Reconstructed Using a Laplacian Vector Field Simulation

Choi

Blemker

2013

PLoS ONE

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Skeletal muscles are characterized by a large diversity in anatomical architecture and function. Muscle force and contraction are generated by contractile fiber cells grouped in fascicle bundles, which transmit the mechanical action between origin and insertion attachments of the muscle. Therefore, an adequate representation of fascicle arrangements in computational models of skeletal muscles is important, especially when investigating three-dimensional muscle deformations in finite element models. However, obtaining high resolution in vivo measurements of fascicle arrangements in skeletal muscles is currently still challenging. This motivated the development of methods in previous studies to generate numerical representations of fascicle trajectories using interpolation templates. Here, we present an alternative approach based on the hypothesis of a rotation and divergence free (Laplacian) vector field behavior which reflects observed physical characteristics of fascicle trajectories. To obtain this representation, the Laplace equation was solved in anatomical reconstructions of skeletal muscle shapes based on medical images using a uniform flux boundary condition on the attachment areas. Fascicle tracts were generated through a robust flux based tracing algorithm. The concept of this approach was demonstrated in two-dimensional synthetic examples of typical skeletal muscle architectures. A detailed evaluation was performed in an example of the anatomical human tibialis anterior muscle which showed an overall agreement with measurements from the literature. The utility and capability of the proposed method was further demonstrated in other anatomical examples of human skeletal muscles with a wide range of muscle shapes and attachment morphologies.

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Section: Methodsmentioning

confidence: 99%

“…A more detailed description can be found in [32]. Because the tracing is performed element by element and is consistent with the given flux values, this method is very robust such that invalid terminations inside the muscle volume or at the muscle surface are avoided.…”

Section: Methodsmentioning

confidence: 99%

Skeletal Muscle Fascicle Arrangements Can Be Reconstructed Using a Laplacian Vector Field Simulation

Choi

Blemker

2013

PLoS ONE

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“…We also note that the use of the discrete divergence map in the Galerkin discretization (21)- (22) is superficial and leads to an identical discrete system. This observation motivates the correction of v in the first term of (27) such that this error vanishes whenever p ∈ P k+1 (E). To this end, we define the correction map Λ E,k : H(div, E) → L 2 (E) such that Λ E,k v is the unique solution to the following optimization problem: (28) min…”

Section: Local Polynomial Correctionsmentioning

confidence: 80%

“…t i−1 · n i−1 t i−1 · n i ϕ i | ei−1 ≡ 0.We again note that these basis functions coincide with the interpolation functions defined in[27] 7. Following the common approach in the polygonal finite element literature, we subdivide the elements into triangles and use the available rules with the desired order of polynomial accuracy.Downloaded 07/18/15 to 128.122.253.228.…”

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confidence: 99%

A Family of $H(div)$ Finite Element Approximations on Polygonal Meshes

Talischi¹

2015

SIAM J. Sci. Comput.

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In this paper, we present a family of H(div)-compatible finite element spaces on strictly convex n-gons, whose construction makes use of generalized barycentric coordinates. In particular, for integers 0 ≤ k ≤ 2, we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order k and whose vector fields have piecewise kth-order polynomial normal traces along the element boundary. These spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and, as such, their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence. We present a general remedy for restoration of optimal convergence that involves "polynomial corrections" of vector fields and their divergence at the element level. These corrections are consistent with one another and require computation of suitable polynomial projection maps. In addition to the theoretical discussion, the performance of the discretization schemes based on the proposed spaces and the accompanying correction maps is numerically evaluated through their application to H(div) eigenvalue problems, mixed and least-squares approximations of linear and nonlinear porous media flow problems. Introduction.Motivated by the challenges of generating meshes for complex and evolving domains, the development of discretization schemes for solving partial differential equations on arbitrary polygonal and polyhedral meshes has become an active area of research in the past decade. The extension of the finite element method in this direction has been primarily focused on investigation and application of H 1 -conforming spaces involving vertex-based elements [37,36,35,22,39]. This relatively narrow scope can be attributed to the availability and ongoing development of the so-called generalized barycentric coordinates which serve as the underlying basis functions of these finite element spaces. The aim of the present work is to make use of barycentric coordinates to construct H(div)-compatible finite element spaces on convex polygonal partitions. These schemes can be employed for solving variational problems involving the H(div) space including porous media flow or mixed formulations of elasticity and Stokes flow. Since H(curl) in two dimensions is isomorphic to H(div) through a rotation, the present work can also be used to obtain discretizations for problems set in H(curl) such as Maxwells equations.A handful of edge-based finite element formulations are available in the literature for polygons. Kuznetsov and Repin [28] proposed a macroelement formulation using Raviart-Thomas spaces on a simplicial partition of the element. Euler, Schuhmann, and Weiland [19] and more recently Gillette, Rand, and Bajaj [23] have explored

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“…The velocity on an interface is assumed to be constant. More sophisticated interpolation schemes like the extended corner-velocity interpolation scheme (ECVI) [13] have earlier been investigated for dG methods for flow diagnostics [28], but will for simplicity not be applied herein.…”

Section: Velocity Interpolationmentioning

confidence: 99%

Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

et al. 2019

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The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-conditioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (i ) a sequential formulation that decouples flow and transport into separate subproblems, and (ii ) a highly efficient Gauss-Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process.This method has been successfully applied to realfield problems using the standard first-order finite vol-Ø.S. Klemetsdal ume discretization. Here, we extend the idea to firstorder dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.

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Velocity interpolation and streamline tracing on irregular geometries

Cited by 45 publications

References 24 publications

Skeletal Muscle Fascicle Arrangements Can Be Reconstructed Using a Laplacian Vector Field Simulation

Skeletal Muscle Fascicle Arrangements Can Be Reconstructed Using a Laplacian Vector Field Simulation

A Family of $H(div)$ Finite Element Approximations on Polygonal Meshes

Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

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