Flux Evaluation in Primal and Dual Boundary-Coupled Problems

Brummelen, van Eh Harald; Zee, van der Kg Kristoffer; Garg, VV; Prudhomme, Serge

doi:10.1115/1.4005187

Cited by 35 publications

(25 citation statements)

References 18 publications

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“…A naive evaluation of traction from the fluid Cauchy stress, − σ 1 n 1 , will converge poorly to the true traction, so we prefer to use variationally-consistent, conservative definitions of traction [68, 122]. In the case of Nitsche’s method, the appropriate discrete traction on surfaces with weakly enforced Dirichlet boundary conditions includes the penalty terms, matching the traction boundary condition of the FSI structural subproblem (11):

t^{h} = - σ_{1}^{h} n_{1} - ρ_{1} {true{true(u_{1}^{h} - û^{h} true) \cdot n_{1} true}}_{-} true(u_{1}^{h} - u_{2} true) + τ_{normalT normalA normalN}^{B} true(true(u_{1}^{h} - u_{2} true) - true(true(u_{1}^{h} - u_{2} true) \cdot n_{1} true) n_{1} true) + τ_{normalN normalO normalR}^{B} true(u_{1}^{h} - u_{2} true) \cdot n_{1} true) n_{1},

where {·} − denotes the negative part of the bracketed quantity, that is, {𝒜} − = 𝒜 if 𝒜 < 0 and {𝒜} − = 0 if 𝒜 ≥ 0.…”

Section: Nitsche’s Methods For Flow Around Immersed Geometriesmentioning

confidence: 99%

An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves

Kamensky

Hsu

Schillinger

et al. 2015

Computer Methods in Applied Mechanics and Engineering

407

414

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In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm. The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid–structure interface traction, arriving at Nitsche’s method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance. To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

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t^{h} = - σ_{1}^{h} n_{1} - ρ_{1} {true{true(u_{1}^{h} - û^{h} true) \cdot n_{1} true}}_{-} true(u_{1}^{h} - u_{2} true) + τ_{normalT normalA normalN}^{B} true(true(u_{1}^{h} - u_{2} true) - true(true(u_{1}^{h} - u_{2} true) \cdot n_{1} true) n_{1} true) + τ_{normalN normalO normalR}^{B} true(u_{1}^{h} - u_{2} true) \cdot n_{1} true) n_{1},

where {·} − denotes the negative part of the bracketed quantity, that is, {𝒜} − = 𝒜 if 𝒜 < 0 and {𝒜} − = 0 if 𝒜 ≥ 0.…”

Section: Nitsche’s Methods For Flow Around Immersed Geometriesmentioning

confidence: 99%

An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves

Kamensky

Hsu

Schillinger

et al. 2015

Computer Methods in Applied Mechanics and Engineering

407

414

View full text Add to dashboard Cite

show abstract

“…We show that a naive formulation of the 'slip' model leads to an ill-posed adjoint problem and adjoint inconsistency [14]. We provide numerical evidence that the corresponding adjoint solution exhibits spurious oscillations on a simple example dealing with a straight channel flow.…”

Section: Introductionmentioning

confidence: 92%

“…However, their work did not consider adjoint techniques and did not use the 'slip' boundary coupling condition. On the other hand, van Brummelen et al [14] and Estep et al [15] have shown the importance of the treatment of boundary flux coupling for the use of adjoint-based techniques. To our best knowledge, no advances in the application of adjoint-based techniques to microfluidics applications, particularly those involving 'slip' boundary coupling, have yet been published in the literature.…”

Section: Introductionmentioning

confidence: 99%

Adjoint-consistent formulations of slip models for coupled electroosmotic flow systems

Garg

Prudhomme

Zee

et al. 2014

Adv. Model. and Simul. in Eng. Sci.

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Background: Models based on the Helmholtz 'slip' approximation are often used for the simulation of electroosmotic flows. The objectives of this paper are to construct adjoint-consistent formulations of such models, and to develop adjoint-based numerical tools for adaptive mesh refinement and parameter sensitivity analysis. Methods: We show that the direct formulation of the 'slip' model is adjoint inconsistent, and leads to an ill-posed adjoint problem. We propose a modified formulation of the coupled 'slip' model, which is shown to be well-posed, and therefore automatically adjoint-consistent. Results: Numerical examples are presented to illustrate the computation and use of the adjoint solution in two-dimensional microfluidics problems. Conclusions: An adjoint-consistent formulation for Helmholtz 'slip' models of electroosmotic flows has been proposed. This formulation provides adjoint solutions that can be reliably used for mesh refinement and sensitivity analysis.

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“…9 shows the drag coefficient C D for flow around a sphere at Re = 100, computed with our immersogeometric method using different B-rep models of the sphere, each with several levels of surface quadrature refinement. The drag force was evaluated using the variationally consistent conservative definition of traction (Bazilevs and Akkerman, 2010;van Brummelen et al, 2011;Xu et al, 2016). The triangulated surfaces cover mesh sizes from 0.08 to 0.005.…”

Section: Flow Around a Spherementioning

confidence: 99%

Direct immersogeometric fluid flow analysis using B-rep CAD models

Hsu

Wang

et al. 2016

Computer Aided Geometric Design

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We present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche's method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the density and distribution of the surface quadrature points are crucial for the weak enforcement of Dirichlet boundary conditions and for obtaining accurate flow solutions. Also, with sufficient levels of surface quadrature element refinement, the quadrature error near the trim curves becomes insignificant. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor directly represented using B-rep. AbstractWe present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche's method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the ...

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Flux Evaluation in Primal and Dual Boundary-Coupled Problems

Cited by 35 publications

References 18 publications

An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves

An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves

Adjoint-consistent formulations of slip models for coupled electroosmotic flow systems

Direct immersogeometric fluid flow analysis using B-rep CAD models

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