Finite element methods for incompressible viscous flow

The paper deals with the rebound of an elastic solid off a rigid wall of a container filled with an incompressible Newtonian fluid. Our study focuses on a collision-free bounce, meaning a rebound without topological contact between the elastic solid and the wall. This has the advantage of omitting any artificial bouncing law. In order to capture the contact-free rebound for very small viscosities an adaptive numerical scheme is introduced.

The here-introduced scheme is based on a Glowinski time scheme and a localized arbitrary Lagrangian–Eulerian map on finite elements in space. The absence of topological contact requires that very thin liquid channels are solved with sufficient accuracy. It is achieved via newly developed geometrically driven adaptive strategies. Using the numerical scheme, we present here a collection of numerical experiments. A rebound is simulated in the absence of topological contacts. Its physical relevance is demonstrated as, with decreasing viscosities, a free rebound in a vacuum is approached. Further, we compare the dynamics with a second numerical scheme; a here-introduced adaptive purely Eulerian level-set method. The scheme produced the same dynamics for large viscosities. However, as it requires a much higher computational cost, small viscosities cannot be reached by this method. The experiments allow for a better understanding of the effect of fluids on the dynamics of elastic objects. Several observations are discussed, such as the amount of elastic and/or kinetic energy loss or the precise connection between the fluid pressure and the rebound of the solid.

The optimal control of thermally convective flows is usually modeled by an optimization problem with constraints of Boussinesq equations that consist of the Navier-Stokes equation and an advection-diffusion equation. This optimal control problem is challenging from both theoretical analysis and algorithmic design perspectives. For example, the nonlinearity and coupling of fluid flows and energy transports prevent direct applications of gradient type algorithms in practice. In this paper, we propose an efficient numerical method to solve this problem based on the operator splitting and optimization techniques. In particular, we employ the Marchuk-Yanenko method leveraged by the $L^2$-projection for the time discretization of the Boussinesq equations so that the Boussinesq equations are decomposed into some easier linear equations without any difficulty in deriving the corresponding adjoint system. Consequently, at each iteration, four easy linear advection-diffusion equations and two degenerated Stokes equations at each time step are needed to be solved for computing a gradient. Then, we apply the Bercovier-Pironneau finite element method for space discretization, and design a BFGS type algorithm for solving the fully discretized optimal control problem. We look into the structure of the problem, and design a meticulous strategy to seek step sizes for the BFGS efficiently. Efficiency of the numerical approach is promisingly validated by the results of some preliminary numerical experiments.

In this work, a stabilized local radial basis function (RBF) collocation method (LRBFCM) is proposed to solve the incompressible Navier–Stokes equations. An improved back ground fictitious grids or meshes technique is proposed to enhance the stability of the LRBFCM. The pressure is computed directly with the coupled velocity field in the Navier–Stokes equations. The explicit time-stepping scheme based on the forward Euler-method is employed for the time discretization, and the free surface of the flow is easy to capture as no numerical integration in the LRBFCM is needed. Numerical results show a high computational speed and a good stability of the proposed LRBFCM.

Natural disasters present ongoing risks to human life and the global economy, with climate change and environmental factors exacerbating these threats. This article introduces an innovative approach to earthquake prediction and modeling, utilizing a combination of modern mathematical techniques, data-driven methodologies, and artificial intelligence (AI). By integrating the finite element method and wavelet collocation method, our approach enables the solution of complex partial differential equations involving fractional derivatives. This integration provides a more accurate prediction of earthquake behavior by leveraging AI analysis of the resulting numerical solutions.

Our advanced modeling technique enables three-dimensional earthquake modeling, surpassing the limitations of traditional methods and offering unprecedented insights. The comprehensive AI approach shows great potential for improving our understanding of earthquake behavior, facilitating the design of earthquake-resistant structures, and effectively mitigating the devastating impacts of earthquakes.

This research significantly contributes to the field of disaster risk reduction and offers practical applications for both researchers and practitioners. By combining mathematical rigor, data-driven insights, and the power of AI, our next-generation approach opens new avenues for enhancing our preparedness and response to seismic events.

In this article we present a numerical method for simulating the sedimentation of balls in a three-dimensional channel filled with an Oldroyd-B fluid. We have combined a distributed Lagrange multiplier/fictitious domain method with a factorization approach from Lozinski and Owens (2003) [22] via an operator splitting technique. The method is validated by comparing obtained results with those reported in literature. Then the study of fluid elasticity on the formation of ball chain (up to six balls) in Oldroyd-B fluids is presented to demonstrate the capability of our methods. For a higher value of elastic number, a longer chain of balls can be formed while settling in an Oldroyd-B fluid as observed experimentally.

We carry out a rigorous error analysis of the first-order semi-discrete (in time) consistent splitting scheme coupled with a generalized scalar auxiliary variable (GSAV) approach for the Navier-Stokes equations with no-slip boundary conditions. The scheme is linear, unconditionally stable, and only requires solving a sequence of Poisson type equations at each time step. By using the build-in unconditional stability of the GSAV approach, we derive optimal global (resp. local) in time error estimates in the two (resp. three) dimensional case for the velocity and pressure approximations.