Papers by E.H. van Brummelen
Computer Methods in Applied Mechanics and Engineering
We present an adaptive simulation framework for binary-fluid flows, based on the Abels-Garcke-Grü... more We present an adaptive simulation framework for binary-fluid flows, based on the Abels-Garcke-Grün Navier-Stokes-Cahn-Hilliard (AGG NSCH) diffuse-interface model. The adaptiverefinement procedure is guided by a two-level hierarchical a-posteriori error estimate, and it effectively resolves the spatial multiscale behavior of the diffuse-interface model. To improve the robustness of the solution procedure and avoid severe time-step restrictions for small-interface thicknesses, we introduce an ε-continuation procedure, in which the diffuse interface thickness (ε) are enlarged on coarse meshes, and the mobility is scaled accordingly. To further accelerate the computations and improve robustness, we apply a modified Backward Euler scheme in the initial stages of the adaptive-refinement procedure in each time step, and a Crank-Nicolson scheme in the final stages of the refinement procedure. To enhance the robustness of the nonlinear solution procedure, we introduce a partitioned solution procedure for the linear tangent problems in Newton's method, based on a decomposition of the NSCH system into its NS and CH subsystems. We conduct a systematic investigation of the conditioning of the monolithic NSCH tangent matrix and of its NS and CH subsystems for a representative 2D model problem. To illustrate the properties of the presented adaptive simulation framework, we present numerical results for a 2D oscillating water droplet suspended in air, and we validate the obtained results versus those of a corresponding sharp-interface model.
Cornell University - arXiv, Jan 4, 2023
In this article, we study the behavior of the Abels-Garcke-Grün Navier-Stokes-Cahn-Hilliard diffu... more In this article, we study the behavior of the Abels-Garcke-Grün Navier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. We consider this so-called sharp-interface limit in the setting of the classical oscillating-droplet problem. To provide reference limit solutions, we derive new analytical expressions for smallamplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. We probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method, in which the refinements are guided by an a-posteriori error-estimation procedure. The adaptive-refinement procedure enables us to consider diffuseinterface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime. For two distinct modes of oscillation, we determine the optimal scaling relation between the diffuse-interface thickness parameter and the mobility parameter. Additionally, we examine the effect of deviations from the optimal scaling of the mobility parameter on the approach of the diffuse-interface solution to the sharp-interface solution.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
The wetting of soft polymer substrates brings in multiple complexities when compared with the wet... more The wetting of soft polymer substrates brings in multiple complexities when compared with the wetting on rigid substrates. The contact angle of the liquid is no longer governed by Young’s Law, but is affected by the substrate’s bulk and surface deformations. On top of that, elastic interfaces exhibit a surface energy that depends on how much they are stretched—a feature known as the Shuttleworth effect (or as surface-elasticity). Here, we present two models through which we explore the wetting of drops in the presence of a strong Shuttleworth effect. The first model is macroscopic in character and consistently accounts for large deformations via a neo-Hookean elasticity. The second model is based on a mesoscopic description of wetting, using a reduced description of the substrate’s elasticity. While the second model is more empirical in terms of the elasticity, it enables a gradient dynamics formulation for soft wetting dynamics. We provide a detailed comparison between the equilibr...
In recent years there has been a growing interest in a particular class of nonlinear parabolic eq... more In recent years there has been a growing interest in a particular class of nonlinear parabolic equations known as phase-field models, which is characterised by evolving diffuse interfaces instead of sharp interfaces. These models are energy dissipative and mass conservative. Moreover, these models exhibit alternating fast and slow variations in time and impose severe demands on the resolution of the interface. There are many semi-implicit finite difference timestepping schemes which inherit these properties at the discrete level [1]. Obviously, to efficiently and accurately simulate these problems, there is a need to employ a posteriori error estimates to drive adaptive mesh refinement and adaptive time-step selection for fully discrete schemes.
Adjoint shape optimization for steady free-surface flows
International Journal for Numerical Methods in Fluids, 2002
ABSTRACT Numerical solution of flows that are partially bounded by a freely moving boundary is of... more ABSTRACT Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. This work examines the suitability of the adjoint shape optimization method for solving steady free-surface flows. It is shown that preconditioning is imperative to avoid mesh-width dependence of the convergence behaviour. Numerical results are presented for 2D flow over an obstacle in a channel. Copyright © 2002 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Fluids, 2005
The numerical solution of fluid-structure interactions with the customary subiteration method inc... more The numerical solution of fluid-structure interactions with the customary subiteration method incurs numerous deficiencies. We propose a novel solution method based on the conjugation of subiteration with a Newton-Krylov method, and demonstrate its superiority and beneficial characteristics.
International Journal for Numerical Methods in Fluids, 2008
We develop a goal-oriented error estimator for finite-element discretizations of fluid-structure-... more We develop a goal-oriented error estimator for finite-element discretizations of fluid-structure-interaction problems. As a model problem, we consider the steady Stokes flow in a 2D channel where part of the channel wall is a flexible segment. We introduce the reference domain approach where the Stokes problem on the variable domain is transformed to a fixed reference-domain. This allows the formulation of a proper dual problem. The dual solution is then used in the evaluation of the error estimate, as usual.
Computer Methods in Applied Mechanics and Engineering, 2003
Numerical simulation of fluid-structure interactions has typically been done using partitioned so... more Numerical simulation of fluid-structure interactions has typically been done using partitioned solution methods. However, partitioned methods are inherently non-conservative and generally numerically unstable. The deficiencies of partitioned methods have motivated the investigation of monolithic solution methods. Conservation is possible for monolithic methods, the conditions have recently been presented in Ref. [1]. In the present paper we investigate the relevance of maintaining conservation for a model fluid-structure interaction problem, viz., the piston problem. To distinguish the effect of the error induced by the interface coupling from the fluid and structure discretization errors, we use fluid subcycling and an exact time-integration method for the structure. A comparison between conservative and non-conservative monolithic methods as well as partitioned methods is made. We show that maintaining conservation has considerable impact on the stability and accuracy of the numerical method. These results also indicate that only for a conservative monolithic scheme the improvement in accuracy over partitioned methods warrants the computational cost associated with a monolithic solution. Moreover, we illustrate the implications that particular combinations of fluid and structure discretizations can have on the conservation properties of the fluid-structure interaction problem.
Computer Methods in Applied Mechanics and Engineering, 2011
We develop duality-based a-posteriori error estimates for functional outputs of solutions of flui... more We develop duality-based a-posteriori error estimates for functional outputs of solutions of fluid-structure-interaction problems. The crucial complication in obtaining these estimates pertains to the derivation of the coupled dual (exact linearizedadjoint) problem owing to the free-boundary character of fluid-structure interaction. We present two approaches to derive the dual problem. In the domain-map linearization approach, the fluid subproblem is first transformed to a fixed reference domain, after which one essentially linearizes with respect to the domain transformation map. In the shape-linearization approach, fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape-derivative techniques. We show that the dual problems correspond to coupled fluid-structure problems with nonstandard coupling conditions. Furthermore, we present numerical experiments that demonstrate the consistency of the dual-based error estimates and their usefulness in goal-oriented adaptive mesh-refinement.
Computational Mechanics, 2012
We present a combined finite-element/boundary-element method to simulate inflation processes, cha... more We present a combined finite-element/boundary-element method to simulate inflation processes, characterized by a light, folded structure enveloping a viscous fluid. The application of the boundary-element method to approximate the flow allows for automatic evolution of the problem domain according to the kinematic condition. Moreover, it provides an intrinsic mechanism to treat the ubiquitous self-contact, common to inflation problems. We numerically verify that self-contact is indeed prevented and demonstrate the versatility and robustness of this method.
Journal of Applied Mechanics, 2011
A crucial aspect in boundary-coupled problems, such as fluid-structure interaction, pertains to t... more A crucial aspect in boundary-coupled problems, such as fluid-structure interaction, pertains to the evaluation of fluxes. In boundary-coupled problems, the flux evaluation appears implicitly in the formulation and consequently, improper flux evaluation can lead to instability. Finite-element approximations of primal and dual problems corresponding to improper formulations can therefore be nonconvergent or display suboptimal convergence rates. In this paper, we consider the main aspects of flux evaluation in finite-element approximations of boundary-coupled problems. Based on a model problem, we consider various formulations and illustrate the implications for corresponding primal and dual problems. In addition, we discuss the extension to free-boundary problems, fluid-structure interaction, and electro-osmosis applications.
SIAM Journal on Numerical Analysis, 2006
Coercivity of the bilinear form in a continuum variational problem is a fundamental property for ... more Coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite-element discretizations: By the classical Lax-Milgram theorem, any conforming discretization of a coercive variational problem is stable; i.e., discrete approximations are well-posed and possess unique solutions, irrespective of the specifics of the underlying approximation space. Based on the prototypical one-dimensional Poisson problem, we establish in this work that most concurrent discontinuous Galerkin formulations for second-order elliptic problems represent instances of a generic conventional formulation and that this generic formulation is noncoercive. Consequently, all conventional discontinuous Galerkin formulations are a fortiori noncoercive, and typically their well-posedness is contingent on approximation-space-dependent stabilization parameters. Moreover, we present a new symmetric nonconventional discontinuous Galerkin formulation based on element Green's functions and the data local to the edges. We show that the new discontinuous Galerkin formulation is coercive on the broken Sobolev space H 1 (P h), viz., the space of functions that are elementwise in the H 1 Sobolev space. The coercivity of the new formulation is supported by calculations of discrete inf-sup constants, and numerical results are presented to illustrate the optimal convergence behavior in the energy-norm and in the L 2 (Ω)-norm.
Applied Numerical Mathematics, 2008
The basic iterative method for solving fluid-structure-interaction problems is a defect-correctio... more The basic iterative method for solving fluid-structure-interaction problems is a defect-correction process based on a partitioning of the underlying operator into a fluid part and a structural part. In the present work we establish for a prototypical model problem that this defect-correction process yields an excellent smoother for multigrid, on account of the relative compactness of the fluid part of the operator with respect to the structural part. We show that the defect-correction process in fact represents an asymptotically-perfect smoother, i.e., the effectiveness of the smoother increases as the mesh is refined. Consequently, on sufficiently fine meshes the fluid-structure-interaction problem can be solved to arbitrary accuracy by one iteration of the defect-correction process followed by a coarse-grid correction. Another important property of the defect-correction process is that it smoothens the error in space/time, so that the coarsening in the multigrid method can be applied in both space and time.
Conservative discretizations of two-uid ow p roblems generally exhibit pressure oscillations. In ... more Conservative discretizations of two-uid ow p roblems generally exhibit pressure oscillations. In this work we show that these pressure oscillations are induced by the loss of a pressure-invariance property under discretization, and we introduce a non-oscillatory conservative method for b a rotropic two-uid ows. The conservative formulation renders the two-uid ow p roblem suitable to treatment by a Godunov-type method. We p resent a modi ed Osher scheme for the two-uid ow p roblem. Numerical results are presented for a translating-interface test case and a shock/interface{collision test case.
Modelling, Analysis and Simulation Riemann-problem and level-set approaches for two-fluid flow computations II. Fixes for solution errors near interfaces
Fixes are presented for the solution errors (`pressure oscillations') that may occur near tw... more Fixes are presented for the solution errors (`pressure oscillations') that may occur near two-fluid interfaces when applying a capturing method. The fixes are analyzed and tested. For two-fluid flows with arbitrarily large density ratios, a variant of the ghost-fluid method appears to be a perfect remedy. Results are presented for compressible water-air flows. The results are promising for a further elaboration of this important application area. The paper contributes to the state-of-the-art in computing two-fluid flows.
Numerical solution of flows that are partially bounded by a freely moving boundary is of great im... more Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. The usual method for solving steady viscous free-surface flow subject to gravitation is alternating time integration of the kinematic condition, and the Navier-Stokes equations with the dynamic conditions imposed, until steady state is reached. This paper shows that at subcritical Froude numbers this time integration approach is necessarily inefficient and proposes an efficient iterative method for solving the steady free-surface flow problem. The new method relies on a different but equivalent formulation of the free-surface flow problem, involving a so-called quasi free-surface condition. The convergence behavior of the new method is shown to be asymptotically mesh width independent. Numerical results are presented for 2D flow over an obstacle in a channel. The results confirm the mesh width independence of the convergence ...
Abstract. Subiteration forms the basic iterative method for solving the aggregated equations in f... more Abstract. Subiteration forms the basic iterative method for solving the aggregated equations in fluid-structure-interaction problems, in which the fluid and structure equations are solved alter-natingly subject to complementary partitions of the interface conditions. In the present work we establish for a prototypical model problem that the subiteration method can be characterized by re-cursion of a nonnormal operator. This implies that the method typically converges nonmonotonously. Despite formal stability, divergence can occur before asymptotic convergence sets in. It is shown that the transient divergence can amplify the initial error by many orders of magnitude, thus inducing a severe degradation in the robustness and efficiency of the subiteration method. Auxiliary results concern the dependence of the stability and convergence of the subiteration method on the physical parameters in the problem and on the computational time step.
and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research ... more and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
Tectonic faults are commonly modelled as Volterra or Somigliana dislocations in an elastic medium... more Tectonic faults are commonly modelled as Volterra or Somigliana dislocations in an elastic medium. Various solution methods exist for this problem. However, the methods used in prac-tice are often limiting, motivated by reasons of computational efficiency rather than geophysical accuracy. A typical geophysical application involves inverse problems for which many different fault configurations need to be examined, each adding to the computational load. In practice, this precludes conventional finite-element methods, which suffer a large computational over-head on account of geometric changes. This paper presents a new non-conforming finite-element method based on weak imposition of the displacement discontinuity. The weak imposition of the discontinuity enables the application of approximation spaces that are independent of the dislocation geometry, thus enabling optimal reuse of computational components. Such reuse of computational components renders finite-element modeling a viable...
A finite-volume method is presented for the computation of compressible flows of two immiscible f... more A finite-volume method is presented for the computation of compressible flows of two immiscible fluids at very different densities. The novel ingredient in the method is a two-fluid linearized Godunov scheme, allowing for flux computations in case of different fluids (e.g., water and air) left and right of a cell face. A level-set technique is employed to distinguish between the two fluids. The level-set equation is incorporated into the system of hyperbolic conservation laws.
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Papers by E.H. van Brummelen