A high-order immersed boundary discontinuous-Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

Brandstetter, Gerd; Govindjee, Sanjay

doi:10.1002/nme.4835

Cited by 18 publications

(21 citation statements)

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“…Elliptic interface problems with discontinuous coefficients across embedded interfaces with corners, giving rise to possibly singular solutions, are considered in a cell merging dG method of Brandstetter and Govindjee [33]. As part of ongoing work (ultimately with the goal of addressing time-dependent interface evolution problems), Kummer [34] considers high-order accurate spatial discretisation for static, time-independent, two-phase incompressible Navier-Stokes equations, using cell agglomeration together with a high-order accurate quadrature method based hierarchical moment fitting [35]. Cell merging is also used by Antonietti et al [36] as part of meshing domains with complex shapes: domain boundaries exhibiting a wide range of geometric scales are meshed with a hierarchy built on refinement in combination with an agglomeration procedure.…”

Section: Background and Motivationmentioning

confidence: 99%

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Saye

2017

Journal of Computational Physics

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Section: Background and Motivationmentioning

confidence: 99%

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Saye

2017

Journal of Computational Physics

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“…Here, we limit ourselves for simplicity of presentation to the purely Dirichlet boundary value problem along Γ. Please refer to for a more general discussion of and alternate boundary conditions. The variational form for reads: Find

normalΦ \in {scriptP}_{s}

, such that

\int_{scriptR} ε_{scriptR} \nabla δ normalΦ \cdot \nabla normalΦ dv + \int_{scriptV} ε_{scriptV} \nabla δ normalΦ \cdot \nabla normalΦ dv = - \int_{{normalΓ}_{B E}} δ normalΦ q_{scriptV} da

for all

δ normalΦ \in {scriptP}_{v}

along with the requirement

normalΦ = \overset{normalΦ}{true}

on Γ.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Here, we limit ourselves for simplicity of presentation to the purely Dirichlet boundary value problem along . Please refer to [30] for a more general discussion of (1) and alternate boundary conditions. The variational form for (1) reads: Findˆ2 P s , such that Z R R rıˆ rˆdv C…”

Section: Introductionmentioning

confidence: 99%

“…With FEBE methods, one needs to deal with singular or nearly singular integrations of Green's function; in the ALE methods, one has to pay special attention to distorted or collapsing elements when modeling the surrounding space. As an alternate, our starting point will be the high‐order discontinuous Galerkin immersed boundary method recently proposed in . This framework has been extened by to the coupled field setting in an Eulerian–Lagrangian formulation including gap closing contact.…”

Section: Introductionmentioning

confidence: 99%

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Cyclic steady states of nonlinear electro-mechanical devices excited at resonance

Brandstetter

Govindjee

2016

Int. J. Numer. Meth. Engng

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SUMMARYWe present an efficient numerical method to solve for cyclic steady states of nonlinear electro-mechanical devices excited at resonance. Many electro-mechanical systems are designed to operate at resonance, where the ramp-up simulation to steady state is computationally very expensive -especially when low damping is present. The proposed method relies on a Newton-Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time-stepping from zero initial conditions. We use a recently developed high-order Eulerian-Lagrangian finite element method in combination with an energy-preserving dynamic contact algorithm in order to solve the coupled electro-mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro-electro-mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed-ups of the form S D 0:6 0:8 , where is the linear damping ratio of the corresponding resonance frequency.

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“…Both approaches rely on planar surface triangulations for the numerical integration of the weak forms, which renders the introduction of quadrature sub-cells inevitable if higher order convergence of the scheme is desired. An interesting alternative has been proposed in [12] where special enrichment functions for common features such as circular sections or corners are introduced. Unfortunately, the presented scheme appears to be limited to second-order accuracy.The development of a DG IBM has first been reported by Fidkowski and Darmofal [13] where the authors study the implicit solution of steady compressible flows over two-dimensional geometries based on an h-adaptive cut cell strategy.…”

mentioning

confidence: 99%

A high‐order discontinuous Galerkin method for compressible flows with immersed boundaries

Müller

Krämer-Eis

Kummer

et al. 2016

Numerical Meth Engineering

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We present a higher order discretization scheme for the compressible Euler and Navier-Stokes equations with immersed boundaries. Our approach makes use of a discontinuous Galerkin discretization in a domain that is implicitly defined by means of a level set function. The zero iso-contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary-fitted cells. In order to retain the full order of convergence of the scheme, it is crucial to perform volume and surface integrals in boundary cells with high accuracy. This is achieved using a linear moment-fitting strategy. Moreover, we apply a non-intrusive cell-agglomeration technique that averts problems with very small and ill-shaped cuts. The robustness, accuracy, and convergence properties of the scheme are assessed in several two-dimensional test cases for the steady compressible Euler and Navier-Stokes equations. Approximation orders range from 0 to 4, even though the approach directly generalizes to even higher orders. In all test cases with a sufficiently smooth solution, the experimental order of convergence matches the expected rate for discontinuous Galerkin schemes. 4 B. MÜLLER ET AL.intersected by the immersed boundary, which allows us to avoid integration sub-cells and, as a result, to extend the scheme to higher approximation orders. Second, we do not require any reformulation of the boundary conditions, which allows us to evade the limitation to adiabatic slip walls. Finally, we have extended the approach to the compressible Navier-Stokes equations. State of the artMethods in the spirit of the original IBM by Peskin [1] have drawn much interest ever since their introduction in 1972 [2]. Within the present work, the term IBM is used in the general sense of methods making use of computational domains that do not conform with the problem domain, hence separating the aspects of discretization and geometry representation to a large extent. This idea is the core of many modern developments in the context of the finite element method such as the extended finite element method (XFEM) [5], the finite cell method [6], Nitsche-type methods [7], and even the finite difference method [8]. In the following, we will focus on numerical methods of the finite element type that are able to deliver higher order convergence rates in the presence of curved immersed problem geometries.First efforts in this direction using XFEM/Nitsche-type approaches have been presented in [9,10] in the context of linear elasticity and in [11] in the context of Stokes flow. Both approaches rely on planar surface triangulations for the numerical integration of the weak forms, which renders the introduction of quadrature sub-cells inevitable if higher order convergence of the scheme is desired. An interesting alternative has been proposed in [12] where special enrichment functions for common features such as circular sections or corners are introduced. Unfortunately, the presented scheme appears to be...

show abstract

A high-order immersed boundary discontinuous-Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

Cited by 18 publications

References 42 publications

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Cyclic steady states of nonlinear electro-mechanical devices excited at resonance

A high‐order discontinuous Galerkin method for compressible flows with immersed boundaries

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