Finite Element Analysis of an Arbitrary Lagrangian–Eulerian Method for Stokes/Parabolic Moving Interface Problem With Jump Coefficients

In this paper, a type of arbitrary Lagrangian–Eulerian (ALE) finite element method in the monolithic frame is developed for a linearized fluid–structure interaction (FSI) problem — an unsteady Stokes/parabolic interface problem with jump coefficients and moving interface, where, the corresponding mixed finite element approximation in both semi- and fully discrete scheme are developed and analyzed based upon one type of ALE formulation and a novel H1- projection technique associated with a moving interface problem, and the stability and optimal convergence properties in the energy norm are obtained for both discretizations to approximate the solution of a transient Stokes/parabolic interface problem that is equipped with a low regularity. Numerical experiments further validate all theoretical results. The developed analytical approaches and numerical implementations can be similarly extended to a realistic FSI problem in the future

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain (DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients. The semi- and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface, where the arbitrary Lagrangian-Eulerian (ALE) technique is employed to deal with the moving and immersed subdomain. Stability and optimal convergence properties are obtained for both schemes. Numerical experiments are carried out for different scenarios of jump coefficients, and all theoretical results are validated

Numerical Analysis of Finite Element Method for a Transient Two-phase Transport Model of Polymer Electrolyte Fuel Cell

AbstractIn this paper, we study a 2D transient two-phase transport model for water species in the cathode gas diffusion layer of hydrogen polymer electrolyte fuel cell (PEFC), the reformulation of water concentration equation is described by using Kirchhoff transformation, and its numerical efficiency is demonstrated by successfully dealing with the discontinuous and degenerate water diffusivity. The semi-discrete and fully discrete finite element approximations with Crank-Nicolson scheme are developed for the present model and the optimal error estimate in H1 norm and the sub-optimal error estimate in L2 norm are established for both finite element schemes

An Efficient Two-grid Method for a Two-phase Mixed-domain Model of Polymer Exchange Membrane Fuel Cell

AbstractIn this paper, an efficientandfast numerical methodis studiedand implementedfora simplifiedtwo-phasemixed domain model of polymer exchange membrane fuel cell (PEMFC), which fully incorporates both the anode and cathode sides, including the conservation equations of mass, momentum, water vapor concentration, liquid water saturationandwater content.Theproposed numericalalgorithmisbasedonthetwo-grid discretization technique,the combined finite element-upwind finitevolume method and some other appropriate linearization schemes. The original nonlinear partial differential equations are only solved on the coarse grid while the fine grid approximation solution is obtained linearly. Therefore the computational time can be reduced tremendously compared with the traditional one-grid method. Numerical experiments of the two-grid method and conventional method for a two-phase mixed domain fuel cell model are carried out, showing that the presented method is effective and accurate for the numerical simulation of PEMFC

A Deep Neural Network/Meshfree Method for Solving Dynamic Two-phase Interface Problems

In this paper, a meshfree method using the deep neural network (DNN) approach is developed for solving two kinds of dynamic two-phase interface problems governed by different dynamic partial differential equations on either side of the stationary interface with the jump and high-contrast coefficients. The first type of two-phase interface problem to be studied is the fluid-fluid (two-phase flow) interface problem modeled by Navier-Stokes equations with high-contrast physical parameters across the interface. The second one belongs to fluid-structure interaction (FSI) problems modeled by Navier-Stokes equations on one side of the interface and the structural equation on the other side of the interface, both the fluid and the structure interact with each other via the kinematic- and the dynamic interface conditions across the interface. The DNN/meshfree method is respectively developed for the above two-phase interface problems by representing solutions of PDEs using the DNNs' structure and reformulating the dynamic interface problem as a least-squares minimization problem based upon a space-time sampling point set. Approximation error analyses are also carried out for each kind of interface problem, which reveals an intrinsic strategy about how to efficiently build a sampling-point training dataset to obtain a more accurate DNNs' approximation. In addition, compared with traditional discretization approaches, the proposed DNN/meshfree method and its error analysis technique can be smoothly extended to many other dynamic interface problems with fixed interfaces. Numerical experiments are conducted to illustrate the accuracies of the proposed DNN/meshfree method for the presented two-phase interface problems. Theoretical results are validated to some extent through three numerical examples

A Projection-Based Time-Segmented Reduced Order Model for Fluid-Structure Interactions

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian--Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (horizontal/vertical) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems

ANALYSIS OF A VIBRATING-BEAM-BASED MICROMIXER

ABSTRACT The mixing of two or more streams in microscale devices is a slowly molecular diffusion process due to the unique laminar flows, and some 'turbulence' based mixing technologies which are effective in macroscales become hard to implement in such small dimensions. The chaotic advection based mixing, depending on the stretching and folding of interface, has been proved to be effective for low Reynolds numbers (Re) and is a very promising technology for micro mixing. We propose a new mixing concept based on a vibrating micro-beam in microfluidic channels to generate chaotic advection to achieve an efficient mixing. The simplicity of the proposed mixer design makes microfabrication process easy for practical applications. The feasibility of the concept is evaluated computationally and moving mesh technique (ALE) is utilized to trace the beam movement. The simulation shows that the mixing quality is determined by parameters such as flow velocities, amplitudes and frequencies of vibrating beam. The Reynolds number (Re) is less than 2.0, Pelect number (Pe) ranges from 5 to 1000, and Strohal number (St) 0.3 to 3.0. It was found that vortex type of flows were generated in microchannel due to the interaction between beam and channel wall. The mixing efficiency with this design is well improved comparing with the flows without beam vibration

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) accompanying with high-contrast physical coefficients across the interface and contacting collisions between the structure and fluidic channel wall when the structure is immersed in the fluid. 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 (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 on the structural surface when the immersed structure contacts the fluidic channel wall. The developed UFMFD-FEM for FSI or 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 are 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.Comment: 32 pages, 42 figures, 5 tables, 66 reference