Finite element analysis of compressible and incompressible fluid-solid systems

This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding a well posed mixed linear eigenvalue problem. Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piecewise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a nonconforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and error estimates independent of the acoustic speed are given. For incompressible fluids, a convenient equivalent stream function formulation and a post-process to compute the pressure are introduced.

     

A Posteriori Error Analysis of a Fully-Mixed Finite Element Method for a Two-Dimensional Fluid-Solid Interaction Problem

In this paper we develop an a posteriori error analysis of a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements, and the fluid is supposed to occupy an annular region surrounding the solid, so that a Robin boundary condition imitating the behavior of the Sommerfeld condition is imposed on its exterior boundary. Dual-mixed approaches are applied in both domains, and the governing equations are employed to eliminate the displacement u of the solid and the pressure $p$ of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart-Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. As the main contribution of this work, we derive a reliable and efficient residual-based a posteriori error estimator for the aforedescribed coupled problem. Some numerical results confirming the properties of the estimator are also reported.     

Analysis of a finite element method for pressure/potential formulation of elastoacoustic spectral problems

A finite element method to approximate the vibration modes of a structure enclosing an acoustic fluid is analyzed. The fluid is described by using simultaneously pressure and displacement potential variables, whereas displacement variables are used for the solid. A mathematical analysis of the continuous spectral problem is given. The problem is discretized on a simplicial mesh by using piecewise constant elements for the pressure and continuous piecewise linear finite elements for the other fields. Error estimates are settled for approximate eigenvalues and eigenfrequencies. Finally, implementation issues are discussed.

     

The finite volume method based on the Crouzeix–Raviart element for a fracture model

In this paper, we present and analyze a finite volume method based on the Crouzeix–Raviart element for the coupled fracture model, where the fluid flow is governed by Darcy's law in the one‐dimensional fracture and two‐dimensional surrounding matrix. In the numerical scheme, the pressure in the matrix and fracture is respectively approximated by the Crouzeix–Raviart elements and piecewise constant functions, and then the velocity is calculated by piecewise constant functions element by element. The existence and uniqueness of the numerical solution are discussed, and optimal order error estimates for both the pressure p and the velocity u are proved on general triangulations. We finally carry out numerical experiments, and results confirm our theoretical analysis.     

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.     

2D simulation of fluid-structure interaction using finite element method

This paper deals with pressure-based finite element analysis of fluid–structure systems considering the coupled fluid and structural dynamics. The present method uses two-dimensional fluid elements and structural line elements for the numerical simulation of the problem. The equations of motion of the fluid, considered inviscid and compressible, are expressed in terms of the pressure variable alone. The solution of the coupled system is accomplished by solving the two systems separately with the interaction effects at the fluid–solid interface enforced by an iterative scheme. Non-divergent pressure and displacement are obtained simultaneously through iterations. The Galerkin weighted residual method-based FE formulation and the iterative solution procedure are explained in detail followed by some numerical examples. Numerical results are compared with the existing solutions to validate the code for sloshing with fluid–structure coupling.     

A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows

This paper proposes a weak Galerkin finite element method to solve incompressible quasi-Newtonian Stokes equations. We use piecewise polynomials of degrees k + 1(k 0) and k for the velocity and pressure in the interior of elements, respectively, and piecewise polynomials of degrees k and k + 1 for the boundary parts of the velocity and pressure, respectively. Wellposedness of the discrete scheme is established. The method yields a globally divergence-free velocity approximation. Optimal priori error estimates are derived for the velocity gradient and pressure approximations. Numerical results are provided to confirm the theoretical results.     

A coupled NMM-SPH method for fluid-structure interaction problems

The main challenges in the numerical simulation of fluid–structure interaction (FSI) problems include the solid fracture, the free surface fluid flow, and the interactions between the solid and the fluid. Aiming to improve the treatment of these issues, a new coupled scheme is developed in this paper. For the solid structure, the Numerical Manifold Method (NMM) is adopted, in which the solid is allowed to change from continuum to discontinuum. The Smoothed Particle Hydrodynamics (SPH) method, which is suitable for free interface flow problem, is used to model the motion of fluids. A contact algorithm is then developed to handle the interaction between NMM elements and SPH particles. Three numerical examples are tested to validate the coupled NMM-SPH method, including the hydrostatic pressure test, dam-break simulation and crack propagation of a gravity dam under hydraulic pressure. Numerical modeling results indicate that the coupled NMM-SPH method can not only simulate the interaction of the solid structure and the fluid as in conventional methods, but also can predict the failure of the solid structure.     

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

Analysis and convergence of a MAC-like scheme for the generalized Stokes problem

We introduce a MAC-like scheme (a covolume method on rectangular grids) for approximating the generalized Stokes problem on an axiparallel domain. Two staggered grids are used in the derivation of the discretization. The velocity is approximated by conforming bilinears over rectangular elements, and the pressure by piecewise constants over macro-rectangular elements. The error in the velocity in the H¹ norm and the pressure in the L² norm are shown to be of first order, provided that the exact velocity is in H² and the exact pressure in H¹, and that the partition family of the domain is regular. © 1997 John Wiley & Sons, Inc.     

Darcy-Stokes问题的统一稳定化有限体积法分析

本文对Darcy-Stokes问题提出了一种统一的稳定化有限体积法.在离散问题中,采用两种剖分,一种为三角形剖分,一种为其对偶四边形剖分.速度及压力分别采用非协调线性元及分片常数元来做逼近.经证明,文中的统一格式,具有稳定性及最优误差估计.最后用数值算例验证了本文的理论结果.     

A multi-mesh finite element method for Lagrange elements of arbitrary degree

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviors of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid–solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier–Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy way to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.     

Accurate pressure post-process of a finite element method for elastoacoustics

Summary. This paper deals with a post-process to obtain a more accurate approximation of the fluid pressure from a finite element computation of the vibration modes of a fluid-structure coupled system. The underlying finite element method, based on a displacement formulation for both media, consists of using Raviart-Thomas elements for the fluid combined with standard continuous elements for the solid. An easy to compute post-process of the pressure is derived. The relation between this post-process and an alternative finite element approximation of the problem based on discretizing the fluid pressure by enriched Crouzeix-Raviart elements is studied. Higher order estimates for the L² norm of the post-processed pressure are proved by exploiting this relation. As a by-product, higher order L² estimates for the solid displacements obtained with the original method are also proved.Member of CIC, Provincia de Buenos Aires, ArgentinaMember of CONICET, Argentina. Partially supported by FONDECYT 7.990.075 and FONDAP in Applied Mathematics, ChilePartially supported by FONDECYT 1.990.346 and FONDAP in Applied Mathematics, Chile     

A weak Galerkin finite element method for a coupled Stokes‐Darcy problem

In this article, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity‐pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. By using the weak Galerkin approach, we consider the two‐dimensional problem with the piecewise constant elements for approximations of the velocity, pressure, and hydraulic head. Stability and optimal error estimates are obtained. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the weak Galerkin approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1352–1373, 2017     

A Nonconforming Finite Element Method of Streamline Diffusion Type for the Incompressible Navier-Stokes Equations

A nonconforming finite element method of streamline diffusion type for solving the stationary and incompressible Navier-Stokes equation is considered. Velocity field and pressure field are approximated by piecewise linear and piecewise constant functions, respectively. The existence of solutions of the discrete problem and the strong convergence of a subsequence of discrete solutions are established. Error estimates are presented for the uniqueness case.     

Contour integral method for European options with jumps

We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.     

Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions

We construct stable, conforming and symmetric finite elements for the mixed formulation of the linear elasticity problem in two dimensions. In our approach we add three divergence-free rational functions to piecewise polynomials to form the stress finite element space. The relation with the elasticity elements and a class of generalized $C^1$ Zienkiewicz elements is also discussed.     

A discontinuous Galerkin method for Stokes equation by divergence-free patch reconstruction

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence-free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle-point problem due to such weak form. The number of degree of freedoms of our method is the same as the number of elements in the mesh for different order of accuracy. The error estimations of the proposed method are given in a classical style, which are then verified by some numerical examples.     

Modified cross-grid finite elements for the stokes problem

The stability of modified cross-grid elements for the approximation of the Stokes problem using continuous piecewise linear polynomials to approximate velocities and piecewise constants to approximate pressures is proved. A key feature of the method is that the mesh for pressure is modified so that the method is stable without augmenting pressure jumps. A numerical test which confirms the stability and the optimal order error estimate is presented.