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1. Error analysis of higher order trace finite element methods for the surface Stokes equation

   https://doi.org/10.1515/jnma-2020-0017  

   Jankuhn, Thomas; Olshanskii, Maxim A.; Reusken, Arnold; Zhiliakov, Alexander (October 2020, Journal of Numerical Mathematics)

   null (Ed.)

   Abstract The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ 3 . The method employs parametric P k - P k −1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere. 

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2. Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

   https://doi.org/10.1002/nme.6158  

   Al Mahbub, Md. Abdullah; He, Xiaoming; Nasu, Nasrin Jahan; Qiu, Changxin; Zheng, Haibiao (July 2019, International Journal for Numerical Methods in Engineering)

   Summary In this paper, we propose and analyze two stabilized mixed finite element methods for the dual‐porosity‐Stokes model, which couples the free flow region and microfracture‐matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual‐porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. 

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3. A PNP ion channel deep learning solver with local neural network and finite element input data

   https://doi.org/10.1088/2632-2153/ad7e7a  

   Lee, Hwi; Chao, Zhen; Cobb, Harris; Liu, Yingjie; Xie, Dexuan (October 2024, Machine Learning: Science and Technology)

   Abstract This paper presents a deep learning method for solving an improved one-dimensional Poisson–Nernst–Planck ion channel (PNPic) model, called the PNPic deep learning solver. The solver combines a novel local neural network, adapted from the neural network with local converging inputs, with an efficient PNPic finite element solver, developed in this work. In particular, the local neural network is extended to handle the complexities of the PNPic model—a system of nonlinear convection–diffusion and elliptic equations with multiple subdomains connected by interface conditions. The PNPic finite element solver efficiently generates input and reference datasets for fast training the local neural network, as well as input datasets for quickly predicting PNPic solutions with high accuracy for a family of PNPic models. Initial numerical tests, involving perturbations of model parameters and interface locations, demonstrate that the PNPic deep learning solver can generate highly accurate numerical solutions. 

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4. A Fluid-Structure Coupled Computational Model for the Certification of Shock-Resistant Elastomer Coatings

   https://doi.org/10.1115/OMAE2020-18501  

   Ma, Wentao; Zhao, Xuning; Wang, Kevin (August 2020, ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering)

   null (Ed.)

   Abstract Shock waves from underwater and air explosions are significant threats to surface and underwater vehicles and structures. Recent studies on the mechanical and thermal properties of various phase-separated elastomers indicate the possibility of applying these materials as a coating to mitigate shock-induced structural failures. To demonstrate this approach and investigate its efficacy, this paper presents a fluid-structure coupled computational model capable of predicting the dynamic response of air-backed bilayer (i.e. elastomer coating – metal substrate) structures submerged in water to hydrostatic and underwater explosion loads. The model couples a three-dimensional multiphase finite volume computational fluid dynamics model with a nonlinear finite element computational solid dynamics model using the FIVER (FInite Volume method with Exact multi-material Riemann solvers) method. The kinematic boundary condition at the fluid-structure interface is enforced using an embedded boundary method that is capable of handling large structural deformation and topological changes. The dynamic interface condition is enforced by formulating and solving local, one-dimensional fluid-solid Riemann problems, which is well-suited for transferring shock and impulsive loads. The capability of this computational model is demonstrated through a numerical investigation of hydrostatic and shock-induced collapse of aluminum tubes with polyurea coating on its inner surface. The thickness of the structure is resolved explicitly by the finite element mesh. The nonlinear material behavior of polyurea is accounted for using a hyper-viscoelastic constitutive model featuring a modified Mooney-Rivlin equation and a stress relaxation function in the form of prony series. Three numerical experiments are conducted to simulate and compare the collapse of the structure in different loading conditions, including a constant pressure, a fluid environment initially in hydrostatic equilibrium, and a two-phase fluid flow created by a near-field underwater explosion. 

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5. An Immersed Finite Element Method for Elliptic Interface Problems with Multi-Domain and Triple Junction Points

   https://doi.org/10.4208/aamm.OA-2018-0175  

   Chen, Yuan; Hou, Songming; Zhang, Xu (October 2019, Advances in applied mathematics and mechanics)

   Interface problems have wide applications in modern scientific research. Obtaining accurate numerical solutions of multi-domain problems involving triple junction conditions remains a significant challenge. In this paper, we develop an efficient finite element method based on non-body-fitting meshes for solving multi-domain elliptic interface problems. We follow the idea of immersed finite element by modifying local basis functions to accommodate interface conditions. We enrich the local finite element space by adding new basis functions for handling non-homogeneous flux jump. The numerical scheme is symmetric and positive definite. Numerical experiments are provided to demonstrate the features of our method. 

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