Schedule for: 18w5077 - Numerical Analysis of Coupled and Multi-Physics Problems with Dynamic Interfaces

In this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition, and derive a well-posed stream function formulation of a class of surface Stokes problems. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. Recently we derived a surface Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. Using this decomposition the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth order equation for the stream function. A particular finite element method for the latter formulation is explained and results of a numerical experiment with this method are presented.

In the talk we discuss a recently introduced finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. Theoretical findings include a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. We show numerical examples that illustrate the theory and demonstrate the practical efficiency of the method.

In this talk, we will explore cell aggregation techniques for finite element numerical approximations with unfitted meshes, when using conforming spaces. We will present and analyze the method, show how it solves the small cut cell problem, and discuss its numerical implementation. We have combined this strategy with octree meshes to deal with large scale problems. We will also show parallel simulations with unfitted methods based on aggregation, and comment on the linear solver and octree handling steps. Finally, we will consider the Stokes problem, and analyze the well-posedness of the aggregated approach when using mixed finite element methods. It will motivate the introduction of a minimal stabilization and the use of new types of extension operators in the aggregated method.

Radial basis functions are highly effective meshfree methods for the solution of PDEs problems. In this talk, we first solve a stationary and evolutionary Stokes problem by global and local divergence free RBFs techniques. We prove that, unlike global techniques, local methods are capable of solving large PDEs Stokes problem in an efficient way. We then, use these RBFs methods to solve null control problems for the Stokes system with few internal scalar controls, which are supported in small sets. We recall, that up to the best of our knowledge, Stokes control problems have not been treated in the literature by radial basis functions (RBFs) methods. The RBFs results are compared with classical mixed finite element methods, proving that the proposed meshfree techniques are competitive and more flexible, due to the lack of mesh, than these classical techniques. We close the talk with some open problems and perspectives in this field. Coauthors: Louis David Breton & Cristhian Montoya

In this talk, we present a fixed mesh method for solving interface inverse problems for an elliptic boundary value problem that is posed in a domain formed with different materials. These inverse problems are to determine the interface that separates different materials by some measurements, either on the boundary and/or in the interior of the domain, about the solution to the boundary problem. We formulate these interface inverse problems as shape optimization problems whose objective functionals depend on the interface. Both the governing partial differential equations and objective functionals are discretized optimally by an immersed finite element (IFE) method on a fixed mesh independent of interface. The formulas for the shape sensitivities of the discretized objective functions are derived within the IFE framework that can be computed accurately and efficiently through the discretized adjoint method. We demonstrate features of this IFE-based shape optimization algorithm by a group of representative interface inverse/design problems.

Nitsche's method can be understood as a stabilized mortar method [1]. In simple situations, a subspace-splitting also leads to a method that avoids computation of the Lagrange multiplier. We discuss generalization of this idea to the elliptic interface problem. [1] R. Stenberg. On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math., 63(1-3):139–148, 1995.

We present a stable finite-element scheme for incompressible flows in time-dependent domains. The time step is independent of the mesh size, and only one linear system is solved on each time step. We consider fluid-structure interaction (FSI) and Navier-Stokes equations in time-dependent domains. The properties of the scheme are shown on several benchmarks and hemodynamic applications. This is the joint work with Maxim Olshanskii (University of Houston), Alexander Danilov, Alexander Lozovskiy and Victoria Salamatova (INM RAS, MIPT). A.Lozovskiy, M.Olshanskii, V.Salamatova, Yu.Vassilevski. An unconditionally stable semi-implicit FSI finite element method. Comput.Methods Appl.Mech.Engrg., V.297, pp.437-454, 2015 A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle. Russian J. Numer. Anal. Math. Modelling, V.32, N4, pp.225-236, 2017 A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain. Comput.Methods Appl.Mech.Engrg., V.333, 55-73, 2018

We present accurate and efficient numerical methods to simulate the deformation of drops and bubbles in Stokes flow with surfactant. Both insoluble and soluble surfactant will be considered. For insoluble surfactant, a new boundary integral (BI) method is presented in which a special quadrature originally developed by Helsing and Ojala is used to retain accuracy for close drop-drop interactions (joint work with Sara Palsson and Anna-Karin Tornberg, KTH). For soluble surfactant, we describe longstanding work and recent results on the development of a hybrid or multiscale BI method that resolves a transition layer of bulk soluble surfactant near the interface (with Jacek Wrobel and Michael Booty, NJIT).

We propose a fully eulerian model for simulating the interaction of fluid-fluid and fluid-solid structures for incompressible fluids and a black-box type numerical scheme for computing its solution. The physical model consists of two Hamilton-Jacobi type PDEs: the fluid-fluid and fluid-solid interfaces are implicitly tracked by a level set equation, and the fluid motion is modeled by the vorticity formulation of Navier-Stokes equation, also of Hamilton-Jacobi type. This approach simplifies the task by allowing us to implement a single numerical scheme for solving both equations. We present the proposed scheme along with several numerical experiments simulating the coalescence of gas bubbles and the oscillations of an elastic membrane that demonstrate the simplicity and robustness of this approach.

In this talk we discuss the finite element approximation of fluid-structure interaction problems. After presenting a finite element version of the Immersed Boundary Method (originally developed by Peskin in the finite difference framework), we show how the introduction of a distributed Lagrange multiplier allows the design of a Fictitious Domain formulation. The properties of our approach include a superior mass conservation, unconditional stability for the time discretization, and rigorous convergence analysis for the steady state solution.

Interface problems arise in several applications including heart models, cochlea models, aquatic animal locomotion, blood cell motion, front-tracking in porous media flows and material science, to name a few. One of the difficulties in these problems is that solutions are normally not smooth across interfaces, and therefore standard numerical methods will lose accuracy near the interface unless the meshes align to it. However, it is advantageous to have meshes that do not align with the interface, especially for time dependent problems where the interface moves with time. Remeshing at every time step can be prohibitively costly, can destroy the structure of the mesh, or can deteriorate the well-conditioning of the stiffness matrix, and affect the stability of the problem. For a simple moving interface fluid-membrane interaction, we present a formal second-order finite element discretization in space and first-order in time where the finite element triangulation does not fit the interface and it is unconditionally stable in time independently of mesh parameters and fluid viscosity and membrane stiffness. This is a joint work with Kyle Dunn and Roger Lui from WPI.

The porosity of a compacting sedimentary basin satisfies a nonlinear pseudo-parabolic partial differential equation. This equation is distinguished from the classical porous medium equation by a third order term, a degenerate elliptic operator in spatial variables acting on the time derivative of the solution. We describe the derivation of the model equation, review classical results for pseudo-parabolic equations and their relation to parabolic equations, present new existence and regularity results for this nonlinear PDE, and show they are consistent with expected behavior of solutions in this context.

Motivated by the modeling of thin structures we study the Stokes problem under punctual forcing. We show the well-posedness of this problem on weighted spaces over general Lipschitz domains. We develop an a posteriori error estimator for this problem and show its reliability and efficiency. Numerical experiments illustrate and complement our theory.

TBD

In this talk, I will introduce a generalized Hele-Shaw problem to model the tumor growth. There are bifurcations arising from the tumor growth model. Some numerical results will be presented for bifurcations and non-trivial solutions on different bifurcation branches. Numerical convergence of boundary integral method of this tumor growth model will also be covered.

In this talk we consider the formulation of FSI problems based on a Fictitious Domain method with a Lagrange multiplier. The time discretization of the resulting system of partial differential equations leads to the solution of a stationary problem with a saddle point structure. We present a rigorous analysis of existence, uniqueness and stability of the solution of the continuous problem and of its finite element discretization. Moreover, we introduce splitting schemes and discuss their stability and convergence properties.

In fluid mechanics the interaction of fluids with distinguishable material properties (e.g. water and air) is referred as multiphase flow. In this work we consider two-phase incompressible flow and concentrate on the representation and time evolution of the interface. There is an extensive list of methods to treat material interfaces. Popular choices include the volume of fluid and level set techniques. We propose a novel level-set like methodology for multiphase flow that preserves the initial mass of each phase. The model combines and reconciles ideas from the volume of fluid and level set methods by solving a non-linear conservation law for a regularized Heaviside of the (distance function) level-set. This guarantees conservation of the volume enclosed by the zero level-set. The equation is regularized by a consistent term that assures a non-singular Jacobian. In addition, the regularization term penalizes deviations from the distance function. The result is a nonlinear monolithic model for a phase conservative level-set where the level-set is given by the distance function. The continuous model is monolithic; meaning that only one equation is needed, doesn’t require any post-processing like: numerical stabilization, re-distancing, artificial compression, flux limiting and others, all of which are commonly used in either level-set or volume of fluid methods. In addition, we have only one parameter that controls the strength of regularization/penalization in the model. We start the presentation reviewing the main ingredients of this model: 1) a conservative level-set method by [Kees et all (2011)], which combines a distanced, non-conservative level-set method with the volume of fluid method via a non-linear correction and 2) elliptic re-distancing by [Basting and Kuzmin(2014)]. Afterwards, we manipulate the conservative level-set method by [Kees et all (2011)] to motivate our formulation. We present a first model which we then modify to resolve some difficulties. Finally, we present a full discretization given by continuous Galerkin Finite Elements in space and a high-order Implicit-Explicit time integration. We demonstrate the behavior of this model by solving different benchmark problems in the literature of level-set methods. Then, we present results of this model coupled with a Navier-Stokes solver to simulate water-air interaction problems in two and three dimensions.

Simulations of fluid and porous media couplings, multi-particles & multi-scale problems are important and challenging because of different governing equations, muilti-scales and multi-connected domain, and complicated interface conditions such as BJ and BJS relations. Most of numerical methods in the literature are based on direct approaches that leads large system of equations, often coupled and ill-conditioned, with lower order accuracy near the interface or boundary. In this talk, we propose a finite difference approach with unfitted meshes. By introducing several augmented variables along the interface, we can decouple the original problem as several Poisson/Helmholtz equations with intermediate jump conditions in the solution and the normal derivatives. One obvious advantage is that a fast Poisson/Helmholtz solver can be utilized. The augmented variables should be chosen such that the Beavers-Joseph-Saffman (BJS) and other interface conditions are satisfied. Another significant strategy is to enforce the divergence condition at the interface from the fluid side. We have shown that the original and transformed systems are equivalent. Because the interface conditions are enforced in strong form, we have observed second order convergence for both of the velocity and the pressure for our constructed non-trivial analytic solutions with circular interfaces. The proposed new method has also been utilized to simulate different flow/porous media setting with complicated interfaces which leads to some interesting simulations results such as effect of corners, orientation effect etc.

We discuss modeling, analysis, and simulation of a free boundary problem arising in one of two applications of current interest. First, we consider methane gas is mixed with water under low temperatures and high pressure conditions, leading to the presence of dissolved gas, free gas, and hydrate crystals. Second, we describe biofilm growth at the poreescale around the grains of porous medium. Depending on the time and spatial scale of interest, the model(s) can be of equilibrium (variational inequality) or kinetic type, and can be described with a phase field model. This is joint work with many students and collaborators to be named in the talk.

We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present unfitted method introduces cell and face unknowns in uncut cells, but doubles the unknowns in the cut cells and on the cut faces. The main difference with classical HHO methods is that a Nitsche-type formulation is used to devise the local reconstruction operator. We prove stability estimates and optimal error estimates in the $H^1$-norm. Robustness with respect to cuts is achieved by a local cell-agglomeration procedure taking full advantage of the fact that HHO methods support polyhedral meshes.

We present stable and convergent mixed finite element methods for the discretisation of poroelasticity and elasticity/poroelasticity coupled equations in a new formulation, where the volumetric contributions to the total stress are merged into an additional unknown. The resulting saddle point formulations can be analysed by means of a Fredholm alternative, after realising that they are compact perturbations of Stokes-like invertible systems. Galerkin schemes are then proposed, and given that specific finite dimensional spaces are chosen adequately, the methods remain stable and optimally convergent in the incompressible limit.

Problems in which immersed elastic structures interact with the surrounding fluid abound in science and engineering. Despite their scientific importance, analysis and numerical analysis of such problems are scarce or non-existent. In this talk, we consider the problem of an elastic filament immersed in a 2D or 3D Stokes fluid. We first discuss our recent results on the analysis of the immersed filament problem in a 2D Stokes fluid (the Peskin problem). We prove well-posedness and immediate regularization of the elastic filament configuration, and discuss the implication of these results for numerical analysis. We will then discuss the immersed filament problem in a 3D Stokes fluid (the Slender Body problem). Here, it has not even been clear what the appropriate mathematical formulation of the problem should be. We propose a mathematical formulation for the Slender Body problem and discuss well-posedness for the stationary version of this problem. Furthermore, we prove that the Slender Body approximation, introduced by Keller and Rubinow in the 1980's and used widely in the fluid-structure interaction community, provides an approximation to the Slender Body problem with some error bound. This is joint work with Analise Rodenberg, Laurel Ohm and Dan Spirn.

The Localized Spectral Decomposition finite element method is based on a hybrid formulation of elliptic partial differential equations, that is then transformed via several space decompositions. Such decompositions make the fomulation embarrassingly parallel and efficient, in particular in the presence of multiscale coefficients. It differs from most of the methods out there since it requires solution's minimum regularity. Also, it is robust with respect to high contrast coefficients.

A novel conservative level set method is introduced. The method builds on recent conservative level set approaches and utilizes an entropy production to construct a balanced artificial diffusion and artificial anti-diffusion. The method is self-tuning, maximum principle preserving, suitable for unstructured meshes, and neither re-initialization of the level set function or reconstruction of the interface is needed for long-time simulation. Computational results in one, two and three dimensions are presented for finite element and finite volume implementations of the method.

Simulations of precipitating convection are usually carried out with cloud resolving models, which typically represent all the different phases of water: water vapor, cloud water, rain water and ice. Here we investigate the question: what is the minimal possible representation of water processes that is sufficient for these models? The simplified models that we present use a Boussinesq approximation, assume fast auto conversion and neglect ice. To test the simplified models, we present simulations of squall lines and scattered convection and show that they qualitatively capture observations made in nature and also seen in more comprehensive cloud resolving models, such as propagation of squall lines with tilted profiles, cold pools, and scattered convection. This is joint work with Andrew J. Majda, Samuel N. Stechmann and Leslie M. Smith.

Many advanced engineering problems require the numerical solution of multidomain, multidimension, multiphysics and multimaterial problems with interfaces. When the interface geometry is highly complex or evolving in time, the generation of conforming meshes may become prohibitively expensive, thereby severely limiting the scope of conventional discretization methods. For instance, the simulation of blood flow dynamics in vessel geometries requires a series of highly non-trivial steps to generate a high quality, full 3D finite element mesh from biomedical image data. Similar challenging and computationally costly preprocessing steps are required to transform geological image data into conforming domain discretizations which respect complex structures such as faults and large scale networks of fractures. Even if an initial mesh is provided, the geometry of the model domain might change substantially in the course of the simulation, as in, e.g., fluid-structure interaction and free surface flow problems, rendering even recent algorithms for moving meshes infeasible. Similar challenges arise in more elaborated optimization problems, e.g. when the shape of the problem domain is subject to the optimization process and the optimization procedure must solve a series of forward problems for different geometric configuration. In this talk, we focus on recent finite element methods on cut meshes (CutFEM) as one possible remedy. CutFEM technologies allow flexible representations of complex or rapidly changing geometries by decomposing the computational domain into several, possibly overlapping domains. Alternatively, complex geometries only described by some surface representation can easily be embedded into a structured background mesh. In the first part of this talk, we briefly review how finite element schemes on cut and composite meshes can be designed by using Nitsche-type imposition of interface and boundary conditions. To make the formulations robust, optimally convergent and to avoid ill-conditioned linear algebra systems, so-called ghost-penalties are added in the vicinity of the boundary and interface. In the second part we demonstrate how CutFEM techniques can be employed to address various challenges from mesh generation to fluid-structure interaction problems, solving PDE systems on embedded manifolds of arbitrary co-dimension and PDE systems posed on and coupled through domains of different topological dimensionality

Weight-adjusted inner products are easily invertible approximations to weighted L2 inner products and mass matrices. These approximations make it possible to formulate very simple time-domain discontinuous Galerkin (DG) discretizations for wave propagation based on the the energy of the system. The resulting methods are low storage, energy stable, and high-order accurate for acoustic and elastic wave propagation in arbitrary heterogeneous media and curvilinear meshes. We conclude with numerical results confirming the stability and high-order accuracy of weight-adjusted DG for acoustic, elastic, and coupled acoustic-elastic waves.

In tracer tests in oil reservoirs, a fluid (water) with a radioactive substance pulse is injected into the porous medium and it is monitoring until its arrival in neighboring extraction wells. From the observations of each extraction well, a tracer breakthrough curve is generated (time vs. tracer concentration). These curves are used to determine the existence of communication channels in underground formations and to characterize the porous medium properties, such as porosity, dispersivity, thickness of the production layer and residual oil saturation, among other applications. In this talk we will present the numerical modeling of the dynamics of the tracers, when the injection/extraction tests are performed in a single-well. We will show results for bipolar flow and a study for the injection phase in partially penetrating wells. Numerical simulations are generated by finite element with bilinear elements. Bibliography [1] J. Cosler: Effect of rate-limited Mass-Transfer on Water Sampling with Partially Penetrating Wells. Ground Water, 42(2) (2004), pp. 203--222 . [2] J.-Sh Chen, Ch.-L. Wu and Ch.-W. Liu: Analysis of contaminant transport towards a partially penetrating extraction well in an anisotropic aquifer. Hydrological Processes, 44 (2010), pp. 2125--2136. [3] J.S. Chen, C.W. Liu. Effect of transverse dispersion on solute transport in a vertical dipole flow test with a tracer. Journal of Hydrology, (2011), pp. 1--11. [4] Coronado M., Sandoval M. L., Escobar-Alfaro G. S. Modeling fluid flow and tracer transport in partially penetrating injection wells. Corrections are submitted to Journal of Petroleum Science and Engineering. March 5, 2018. [5] Sandoval M. L., Coronado M., Grande-Sánchez S. Dynamics of vertical dipole tracer tests in a sandy-clay reservoir. Paper in preparation.

We develop exact polynomial sequences on Alfeld splits in any spatial dimension and for any polynomial degree. An Alfeld split of a tetrahedron is obtained by connecting the vertices of an $n$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces. This is joint work with Guosheng Fu and Johnny Guzmán.