Algebraic distance estimations for enriched isogeometric analysis

Upreti, Kritika; Song, Tao; Tambat, A.; Subbarayan, Ganesh

doi:10.1016/j.cma.2014.07.012

Cited by 16 publications

(33 citation statements)

References 26 publications

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“…(44)). The time complexity of computing the algebraic distance field was analyzed in [25]. Now, consider a Bezier curve of degree p, the time complexity of constructing algebraic distance field is 2 3 p 3 + O( p 2 ).…”

Section: Proofmentioning

confidence: 99%

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A sharp interface isogeometric solution to the Stefan problem

Song

Upreti

Subbarayan

2015

Computer Methods in Applied Mechanics and Engineering

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Section: Proofmentioning

confidence: 99%

“…Upreti et al [25] discussed in detail the computation of algebraic distance estimates from curves and surfaces. The main idea is to "implicitize" the parametric entity and use the level set of implicitized function as a measure of distance.…”

Section: Review Of Algebraic Distance Fieldsmentioning

confidence: 99%

A sharp interface isogeometric solution to the Stefan problem

Song

Upreti

Subbarayan

2015

Computer Methods in Applied Mechanics and Engineering

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“…This is because the behavioral influence of the immersed boundaries decays monotonically with distance. While traditionally distances are estimated using Newton-Raphson iterations, recently, Upreti et al [23,24] developed algebraic procedures for efficient computation of distance estimates from curves and surfaces. The method developed by Upreti et al relies on converting the parametric NURBS geometry to its implicit form using the Dixon resultant, and constructing level sets on the implicit form of the geometry.…”

Section: Background On Algebraic Level Setsmentioning

confidence: 99%

Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces

et al. 2020

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Point projection is an important geometric need when boundaries described by parametric curves and surfaces are immersed in domains. In problems where an immersed parametric boundary evolves with time as in solidification or fracture analysis, the projection from a point in the domain to the boundary is necessary to determine the interaction of the moving boundary with the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge, we present in this paper an algebraic procedure to project a point on to Non-uniform rational B-spline (NURBS) curves and surfaces. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bézier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain are then used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G 0 continuity—problems where the Newton–Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively.

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“…This can be accomplished either with numerical signed distance computations 8,42,43 or analytic implicitisation techniques. 44 The use of an implicit instead of a parametric representation greatly simplifies all geometric operations. For instance, the classification of elements as solid, fluid, or cut involves only the evaluation of the signed distance function (x) (see Figure 4).…”

Section: Implicit Geometry Descriptionmentioning

confidence: 99%

An immersed discontinuous Galerkin method for compressible Navier‐Stokes equations on unstructured meshes

Hong

Febrianto

Zhang

et al. 2019

Numerical Methods in Fluids

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Summary We introduce an immersed high‐order discontinuous Galerkin method for solving the compressible Navier‐Stokes equations on non–boundary‐fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are advanced in time with an explicit time marching scheme. The discretisation meshes may contain simplicial (triangular or tetrahedral) elements of different sizes and need not be structured. On the discretisation mesh, the fluid domain boundary is represented with an implicit signed distance function. The cut‐elements partially covered by the solid domain are integrated after tessellation with the marching triangle or tetrahedra algorithms. Two alternative techniques are introduced to overcome the excessive stable time step restrictions imposed by cut‐elements. In the first approach, the cut‐basis functions are replaced with the extrapolated basis functions from the nearest largest element. In the second approach, the cut‐basis functions are simply scaled proportionally to the fraction of the cut‐element covered by the solid. To achieve high‐order accuracy, additional nodes are introduced on the element faces abutting the solid boundary. Subsequently, the faces are curved by projecting the introduced nodes to the boundary. The proposed approach is verified and validated with several two‐ and three‐dimensional subsonic and hypersonic low Reynolds number flow applications, including the flow over a cylinder, a space capsule, and an aerospace vehicle.

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Algebraic distance estimations for enriched isogeometric analysis

Cited by 16 publications

References 26 publications

A sharp interface isogeometric solution to the Stefan problem

A sharp interface isogeometric solution to the Stefan problem

Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces

An immersed discontinuous Galerkin method for compressible Navier‐Stokes equations on unstructured meshes

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