We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)]. This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ.33 (2008) 1018–1032], derived for the case of plates when $S\subset {ℝ}^2$. The convergence holds provided...

We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)]. This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ.33 (2008) 1018–1032], derived for the case of plates when $S\subset {ℝ}^2$. The convergence holds provided...

We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells...

We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with...

A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is...

The operator $L_0:D_{L_0}\subset H\to H$, $L_{0}u=\frac{1}{r}\frac{d}{dr}\left\{r\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)\right]\right\}$, $D_{L_0}=\{u\in C^4\left([0,R]\right),u^{\text{'}}\left(0\right)=u^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)=0,u\left(R\right)=u^{\text{'}}\left(R\right)=0\}$, $H=L_{2,r}(0,R)$ is shown to be essentially self-adjoint, positive definite with a compact resolvent. The conditions on $L_0$ (in fact, on a general symmetric operator) are given so as to justify the application of the Fourier method for solving the problems of the types $L_{0}u=g$ and $u_{tt}+L_{0}u=g$, respectively.

In this paper we present a unified approach to obtain integral representation formulas for describing the propagation of bending waves in infinite plates. The general anisotropic case is included and both new and well-known formulas are obtained in special cases (e.g. the classical Boussinesq formula). The formulas we have derived have been compared with experimental data and the coincidence is very good in all cases.

The author introduces a global measure of initial deflection given by the energy norm. Solving the formulated minimization problem with a subsidiary condition the most dangerous initial deflection shape is obtained. The theoretical results include a wide range of stability type structural problems.

We consider the singular boundary value problem $${\left(t^n{u}^{\text{'}}\left(t\right)\right)}^{\text{'}}+t^{n}f(t,u\left(t\right))=0,\underset{t\to 0+}{lim}{t}^n{u}^{\text{'}}\left(t\right)=0,a_{0}u\left(1\right)+a_1{u}^{\text{'}}(1-)=A,$$ where $f(t,x)$ is a given continuous function defined on the set $(0,1]\times (0,\infty )$ which can have a time singularity at $t=0$ and a space singularity at $x=0$. Moreover, $n\in \mathbb{N}$, $n\ge 2$, and $a_0$, $a_1$, $A$ are real constants such that $a_0\in (0,\infty )$, whereas $a_1,A\in [0,\infty )$. The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in...

A unilateral boundary-value condition at the left end of a simply supported rod is considered. Variational and (equivalent) classical formulations are introduced and all solutions to the classical problem are calculated in an explicit form. Formulas for the energies corresponding to the solutions are also given. The problem is solved and energies of the solutions are compared in the pertubed as well as the unperturbed cases.