Moving Interface Problems Research Articles

A unified-field monolithic fictitious domain-finite element method for fluid-structure-contact interactions and applications to deterministic lateral displacement problems

Based upon two overlapped, body-unfitted meshes, a type of unified-field monolithic fictitious domain-finite element method (UFMFD-FEM) is developed in this paper for moving interface problems of dynamic fluid-structure interactions (FSI), which is accompanied with high-contrast physical coefficients across the interface and with contacting collisions between the structure and fluidic channel wall in the immersed FSI case. In particular, the proposed novel numerical method consists of a monolithic, stabilized mixed finite element method within the frame of fictitious domain/immersed boundary method (FD/IBM) for generic fluid-structure-contact interaction (FSCI) problems in the Eulerian–updated Lagrangian description, while involving the no-slip type of interface conditions on the fluid-structure interface, and the repulsive contact force/stress on the structural surface when the immersed structure contacts the fluidic channel wall. The developed UFMFD-FEM for FSI/FSCI problems can deal with the structural motion with large rotational and translational displacements and/or large deformation in an accurate and efficient fashion, which is first validated by two benchmark FSI problems and one FSCI model problem, then by experimental results of a realistic FSCI scenario – the microfluidic deterministic lateral displacement (DLD) problem that is applied to isolate circulating tumor cells (CTCs) from blood cells in the blood fluid through a cascaded filter DLD microchip in practice, where a particulate fluid with the pillar obstacles effect in the fluidic channel, i.e., the effects of fluid-structure interaction and structure collision, play significant roles to sort particles (cells) of different sizes with tilted pillar arrays. The developed unified-field, monolithic fictitious domain-based mixed finite element method can be seamlessly extended to more sophisticated, high dimensional FSCI problems with contacting collisions between the moving elastic structure and fluidic channel wall.

The random feature method for solving interface problems

Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along another line, mesh-free methods bypass mesh generation but lack robustness in terms of convergence and accuracy due to the low regularity of solutions. In this study, we propose a novel method for solving interface problems within the framework of the random feature method (RFM). This approach utilizes random feature functions in conjunction with a partition of unity as approximation functions, and solves a linear least-squares system to obtain the approximate solution. In the context of interface problems, two innovative and crucial components are incorporated into the RFM. Firstly, we utilize two sets of random feature functions on each side of the interface, allowing for the inclusion of low regularity or even discontinuous behaviors in the solution. Secondly, the construction of the loss function is based on the assessment of the partial differential equation, initial/boundary conditions, and the interface condition on collocation points. This approach ensures that these conditions are equally satisfied. Consequently, the challenges arising from geometric complexity primarily manifest in the generation of collocation points, a task amenable to standard methods. Importantly, the proposed method retains its meshfree characteristics and robustness when addressing problems featuring intricate geometries. We validate our method through a series of linear interface problems with increasingly complex geometries, including two-dimensional elliptic and three-dimensional Stokes interface problems, a three-dimensional elasticity interface problem, a moving interface problem with topological change, a dynamic interface problem with large deformation, and a linear fluid–solid interaction problem with complex geometry. Our findings show that despite the solution often being only continuous or even discontinuous, our method not only eliminates the need for mesh generation but also maintains high accuracy, akin to the spectral collocation method for smooth solutions. Remarkably, for the same accuracy requirement, our method requires two to three orders of magnitude fewer degrees of freedom than traditional methods, demonstrating its significant potential for solving interface problems with complex geometries and predetermined intricate evolution.

Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data: gd(t,x), gfr(t,x), U0(x) and ζ0(x), we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space Wp1,2(Q)×Wp1,2(Σ). Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation.

Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers

The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.

A level-set immersed boundary method for reactive transport in complex topologies with moving interfaces

A simulation framework based on the level-set and the immersed boundary methods (LS-IBM) has been developed for reactive transport problems in porous media involving a moving solid-fluid interface. The interface movement due to surface reactions is tracked by the level-set method, while the immersed boundary method captures the momentum and mass transport at the interface. The proposed method is capable of accurately modeling transport near evolving boundaries in Cartesian grids. The framework formulation guarantees second order accuracy in space. Since the interface velocity is only defined at the moving boundary, an interface velocity propagation method is also proposed. The method can be applied to other moving interface problems of the “Stefan” type. Here, we validate the proposed LS-IBM both for flow and transport close to an immersed object with reactive boundaries as well as for crystal growth. The proposed method provides a powerful tool to model more realistic problems involving moving reactive interfaces in complex domains.

A qualitative analysis of a second-order anisotropic phase-field transition system endowed with a general class of nonlinear dynamic boundary conditions

The paper is concerned with the study of a nonlinear second-order anisotropic phase-field transition system of Caginalp type, subject to nonlinear and in-homogeneous dynamic boundary conditions (in both unknown functions). Under certain hypothesis on the input data: $ f_{_1}(t,x) $, $ f_{_2}(t,x) $, $ w_{_1}(t,x) $, $ w_{_2}(t,x) $, $ u_0(x) $, $ \alpha_0(x) $, $ \varphi_0(x) $ and $ \xi_0(x) $, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $ W^{1,2}_p(Q)\times W^{1,2}_p(\Sigma) $, $ W^{1,2}_\nu(Q)\times W^{1,2}_p(\Sigma) $. Here we extend the previous results concerned with nonlinearity of cubic type, allowing to the present mathematical model to be more capable for describing the complexity of a wide class of real physical phenomena (moving interface problems, image processing, the phase changes at the boundary $ \partial\Omega $, etc.).

High-order unfitted characteristic finite element methods for moving interface problem of Oseen equations

Moving interface problems have broad applications in multi-phase flows and fluid–structure interactions. The design of high-order numerical methods for such problems becomes challenging, in particular, when the interface is severely deformed. We propose kth-order unfitted characteristic finite element methods (k=2,3,4) for solving two-dimensional Oseen equations with a time-varying interface. In each time step, the interface is not specified with an explicit function, but constructed numerically with a (k+1)th-order interface-tracking algorithm. We obtain optimal convergence orders of both the discrete velocity and the discrete pressure under the H1-norm. Our convergence analysis includes errors of the approximate interface and the time integration and spatial discretization of the governing equations. Numerical experiments are performed on a rotating object and a severely deformed interface and show optimal convergence orders of the method.

Numerical aspects of non-Newtonian droplet dynamics using a free surface flow framework

<h2>Abstract</h2> In this talk we present numerical progresses for solving non-Newtonian free surface flows involving droplet dynamics. We revisit the construction of the front-tracking/marker-and-cell based schemes to deal with moving interface problems. The main purpose of our simulations is to investigate the combined effects of the viscoelasticity, plasticity, thixotropy, surface tension and gravity on the impact of droplets. In addition, we present some recent results in the collision dynamics of viscoelastic droplets. Finally, we describe a machine-learning based strategy for computing the curvature of the moving interface in the context of the front-tracking methodology.

Corrected finite element methods on unfitted meshes for Stokes moving interface problem

We propose an explicit in time corrected finite element method for a problem with a moving elastic interface in Stokes flow, using unfitted meshes and standard finite element approximation spaces. Piecewise polynomial correction functions are computed to accurately capture the discontinuities of the solution resulting from the elastic force applied to the fluid and are used to correct the load vector. The method is proven to be optimal for the steady state problem. Additionally, we propose a trigonometric interpolant to approximate and evolve the interface and explicit Euler scheme to advance in time. Numerical examples demonstrate the third and second order accuracy of the method for the velocity and pressure, respectively, and the stability of the numerical scheme.

An asymptotic analysis and numerical simulation of a prostate tumor growth model via the generalized moving least squares approximation combined with semi-implicit time integration

This paper focuses on presenting an asymptotic analysis and employing simulations of a model describing the growth of local prostate tumor (known as a moving interface problem) in two-dimensional spaces. We first demonstrate that the proposed mathematical model produces the diffuse interfaces with the hyperbolic tangent profile using an asymptotic analysis. Next, we apply a meshless method, namely the generalized moving least squares approximation for discretizing the model in the space variables. The main benefits of the developed meshless method are: First, it does not need a background mesh (or triangulation) for constructing the approximation. Second, the obtained numerical results suggest that the proposed meshless method does not need to be combined with any adaptive technique for capturing the evolving interface between the tumor and the neighboring healthy tissue with proper accuracy. A semi-implicit backward differentiation formula of order 1 is used to deal with the time variable. The resulting fully discrete scheme obtained here is a linear system of algebraic equations per time step solved by an iterative scheme based on a Krylov subspace, namely the generalized minimal residual method with zero–fill incomplete lower–upper preconditioner. Finally, some simulation results based on estimated and experimental data taken from the literature are presented to show the process of prostate tumor growth in two-dimensional tissues, i.e., a square, a circle and non-convex domains using both uniform and quasi-uniform points.

Stefan problem for a nonergodic facilitated exclusion process

We consider the facilitated exclusion process, which is a nonergodic, kinetically constrained exclusion process. We show that in the hydrodynamic limit, its macroscopic behavior is governed by a free boundary problem. The particles evolve on the one-dimensional lattice according to jump rates which are degenerate, since they can vanish on non-trivial configurations and create distinct phases: indeed, configurations can be totally blocked (they cannot evolve under the dynamics), ergodic (they belong to an irreducible component), or transient (after a transitive period of time they will become either blocked or ergodic). We additionally prove that the microscopic separation into blocked/ergodic phases fully coincides with the moving interface problem given by the hydrodynamic equation.

Element-Free Galerkin Method Modeling of Thermo-Elastic-Plastic Behavior for Continuous Casting Round Billet

When solving the coupled thermal-mechanical problem of the metal solidification, the mesh-based numerical methods suffer from the inconsistency of mesh lines and curved boundaries. Furthermore, the mesh should be continually reconstructed to comply with the interface when dealing with the moving boundary such as phase transition, which brings great difficulties to the numerical calculation. Based on the plane stress assumption, the present work establishes a two-dimensional thermo-elastic-plastic calculation model for continuous casting round billets using the Element-Free Galerkin method, and the correction scheme of elastoplastic transition region has been explored. The results show that the correction scheme has a slight effect on magnitude of the equivalent stress and no impact on the subsequent plastic deformation zone. The elastoplastic transition region of shell surface appears in the range of 100 to 200 mm below the meniscus, and the averaged plastic stress is about 2.27 times of the elastic stress at the mold outlet. Near the interface region of liquid–solid phase exhibits an elastic deformation behavior, while far from the interface region appears plastic deformation in which the elastic stress remains invariable. The Element-Free Galerkin method is discretized by a series of arbitrary nodes, which exhibits high accuracy and great flexibility in dealing with moving interface problems. It provides a powerful approach to calculate and analyze the heat transfer/mechanical behavior with complex geometric boundaries such as round billet, chamfered billet, and beam blank.

The local tangential lifting method for moving interface problems on surfaces with applications

In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces. One novelty of the new method is the local tangential lifting method to construct discrete delta functions on surfaces. The idea of the local tangential lifting method is to transform a local surface problem to a local two dimensional one on the tangent planes of surfaces at some selected surface nodes. Moreover, a surface version of the front tracking method is developed to track moving interfaces on surfaces. Strategies have been developed for computing geodesic curvatures of interfaces on surfaces. Various numerical examples are presented to demonstrate the accuracy of the new methods. It is also interesting to see the comparison of the dendritic solidification processes in two dimensional spaces and on surfaces.

Robust modelling of implicit interfaces by the scaled boundary finite element method

In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is O(h2) in L2 norm and O(h) in the H1 semi-norm. Further the condition number increases with the mesh size similar to the FEM.

The Fictitious Domain Method with Sharp Interface for Elasticity Systems with General Jump Embedded Boundary Conditions

In the framework of the fictitious domain methods with immersed interfaces for the elasticity problem, the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method. The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain. Here, we present a cell-centered finite volume method to discretize the fictitious domain problem. The proposed method is numerically validated for different test cases. This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.

Solving Parabolic Moving Interface Problems with Dynamical Immersed Spaces on Unfitted Meshes: Fully Discrete Analysis

Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid a mesh regeneratio...

A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

&lt;p style='text-indent:20px;'&gt;In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.&lt;/p&gt;

Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains.

A shape optimization method for moving interface problems governed by the heat equation

The one dimensional Stefan problem is reformulated as a shape optimization problem for the position of the phase transition as a function of time. The functional to be minimized is the mismatch of the Dirichlet to Neumann map at the moving interface. We show that the minimizer is the only stationary point of the shape functional. A gradient based optimization method is derived using shape calculus. The state and adjoint equations of the heat equation are solved with integral equation techniques which avoid a discretization in the domain. A Nyström quadrature method is analyzed and numerical results are presented.

Upscaling of evolving microstructures in a porous medium

Crystal dissolution and precipitation problems often originate in the fields of material sciences, chemical engineering, soil mechanics, hydrology, and other related fields. In this paper, we consider dissolution and precipitation of immobile species (minerals) on the surface of the solid parts and diffusion and reaction of mobile species in the pore space of a porous medium. The mathematical modeling of such processes leads to a system of semilinear parabolic partial differential equations coupled with a system of nonlinear ordinary differential equations, which involves multi-valued right-hand sides. The geometry under consideration here is evolving due to dissolution and precipitation on the interfaces. This moving interface problem is dealt with the help of a level set function. The pore (micro) scale model is upscaled to the macroscale by periodic homogenization and asymptotic expansion techniques.