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A FEM-DtN formulation for a nonlinear exterior problem in incompressible elasticity
Abstract
this paper we combine the usual finite element method (FEM) with a Dirichlet-to-Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in nonlinear incompressible 2d-elasticity. We show that the variational formulation can be written in a Stokes-type mixed form with a linear constraint and a nonlinear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, an a-posteriori error estimate, well suited for adaptive computations, is also given.
... The coupling of finite elements and boundary elements has developed into a very powerful method to tackle a large class of transmission problems in physics and engineering sciences (see, e.g.,123456). In recent years an increasing interest evolved to apply mixed methods instead of usual finite-element methods together with either boundary integral equations or Dirichletto-Neumann mappings (see, e.g.,789). Often in applications a mixed finite-element method is more beneficial than the standard FEM, e.g., in structural mechanics via mixed methods stresses are computed more accurately than displacements. ...
We analyze a least squares formulation for the numerical solution of second-order lineartransmission problems in two and three dimensions, which allow jumps on the interface. In a bounded domain the second-order partial differential equation is rewritten as a first-order system; the part of the transmission problem which corresponds to the unbounded exterior domain is reformulated by means of boundary integral equations on the interface. The least squares functional is given in terms of Sobolev norms of order -1 and of order 1/2. These norms are computed by approximating the corresponding inner products using multilevel preconditioners for a second-order elliptic problem in a bounded domain Ω and for the weakly singular integral operator of the single layer potential on its boundary ∂Ω. As preconditioners we use both multigrid and BPX algorithms, and the preconditioned system has bounded or mildly growing condition number. Numerical experiments confirm our theoretical results.
This paper deals with a least-squares formulation of a second order transmission problem for linear elasticity. The problem in the unbounded exterior domain is rewritten with boundary integral equations on the boundary of the inner domain. In the interior domain we treat a linear elastic material which can also be nearly incompressible. The least-squares functional is given in terms of the H˜−1(Ω) and H1/2(Γ)H1/2(Γ) norms. These norms are realized by solution operators of corresponding dual norm problems which are approximated using multilevel preconditioners.
A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.
We extend the applicability of stable mixed finite elements for linear plane elasticity, such as PEERS, to a mixed variational formulation of hyperelasticity. The present approach is based on the introduction of the strain tensor as a further unknown, which yields a two-fold saddle point nonlinear operator equation for the corresponding weak formulation. We provide the uniqueness of solution for the continuous and discrete schemes, and derive the usual Cea estimate for the associated error. Finally, a reliable a-posteriori error estimate, based on the solution of local Dirichlet problems, and well suited for adaptive computations. is also given.
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