Least-Squares Finite Element Formulation for Fluid-Structure Interaction

Fluid-structure interaction problems prove difficult due to the coupling between fluid and solid behavior. Typically, different theoretical formulations and numerical methods are used to solve fluid and structural problems separately. The least-squares finite element method is capable of accurately solving both fluid and structural problems. This capability allows for a simultaneously coupled fluid structure interaction formulation using a single variational approach to solve complex and nonlinear aeroelasticity problems. The least-squares finite element method was compared to commonly used methods for both structures and fluids individually. The fluid analysis was compared to finite differencing methods and the structural analysis type compared to traditional Weak Galerkin finite element methods. The simultaneous solution method was then applied to aeroelasticity problems with a known solution. Achieving these results required unique iterative methods to balance each domain\u27s or differential equation\u27s weighting factor within the simultaneous solution scheme. The scheme required more computational time but it did provide the first hands-off method capable of solving complex fluid-structure interaction problems using a simultaneous least-squares formulation. A sequential scheme was also examined for coupled problems

Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems

We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations. We can compute branches of solutions with limit points, bifurcation points, etc. Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations

High-accuracy finite-element methods for positive symmetric systems

AbstractA nonstandard-type “least'squares” finite-element method is proposed for the solution of first-order positive symmetric systems. This method gives optimal accuracy in a norm similar to the H1 norm. When a regularity condition holds it is optimal in L2 as well. Otherwise, it gives errors suboptimal by only h12 (where h is the mesh diameter). Thus, it has greater accuracy than usual finite-element, finite-difference or least-squares methods for such problems. In addition, the spectral condition number of the associated linear system is only O(h−1) vs. O(h−2) for the usual least-squares methods.Thus, the method promises to be an efficient, high-accuracy method for hyperbolic systems such as Maxwell's equations. It is also equally promising for mixed-type equations that have a formulation as a positive symmetric system

Petrov-Galerkin and Galerkin/Least Squares stabilized approaches for advectivediffusive transport problems

[Abstract] The Finite Element method with a Galerkin type weighting is a straight-forward weighted residual method that has been sucessfuly used in many engineering applications, specially in Solid Mechanics. However, this method yields oscillatory solutions when it is applied to high-advective problems in Fluid Mechanics. Several stabilized numerical formulations have been proposed in the last years to overcome these inestabilities. The common methodology of most of these approaches is based on the addition of a term to the Galerkin formulation, in order to enhance the estability behaviour while preserving the weighting residual scheme. In this paper, we focus our attention in the Stream Upwind/Petrov Galerkin method (SUPG), and the Galerkin/least-squares method (GLS). We will review the mathematical formulation of both of them, as well as the key concept of their respective fundamentals and derivations, i.e. the exact artificial diffusion method for the SUPG and the Least Squares Finite Element method for the GLS. Finally, we will present a comparision between both methods, pointing out important coincidences and estabishing their mutual relations.Xunta de Galicia; PGDIT99MAR1180

Numerical identification of a variable parameter in 2d elliptic boundary value problem by extragradient methods

This work focuses on the inverse problem of identifying a variable parameter in a 2-D scalar elliptic boundary value problem. It is well-known that this inverse problem is highly ill-posed and regularization is necessary for its stable solution. The inverse problem is studied in an optimization framework, which is the most suitable framework for incorporating regularization. This optimization problem is a constrained optimization problem where the constraint set is a closed and convex set of the admissible coefficients. As an objective functional, we use both the output least squares and modified output least squares functionals. It is known that the most commonly used iterative schemes for such problems require strong monotonicity of the objective functionals derivative. In the context of the considered inverse problem, this is a very stringent requirement and is achieved through a careful selection of the regularization parameter. In contrast, extragradient type methods only require the derivative of the objective functional to be monotone and this allows a greater flexibility for the selection of the regularization parameter. In this work, we use the finite element method for the discretization of the inverse problem and apply the most commonly studied extragradient methods

A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation

Discretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(hk+1/2) error estimate in the L2-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization

A Pressure-Robust Discretization of Oseen's Equation Using Stabilization in the Vorticity Equation

Discretization of Navier--Stokes equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressure-independent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an $O(h^{k+\frac12})$ error estimate in the $L^2$-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based streamline upwind Petrov--Galerkin stabilization