Asymptotic Analysis - Volume 102, issue 3-4

Authors: Shubov, Marianna A.

Article Type: Research Article

Abstract: The present paper is the second one in a series of three works dealing with a mathematical model for an electro-mechanical energy harvester. The harvester is designed as a beam with a piezoceramic layer attached to its top face. A pair of conductive electrodes, covering the top and bottom faces of the piezoceramic layer, are connected to a resistive load. The model is governed by a system of two differential equations. The first of them is the equation of the Euler–Bernoulli beam model subject to actions of an external damping and of an external force. The second equation represents the …Kirchhoff’s law for the electric circuit. Both equations are coupled by means of the direct and converse piezoelectric effects. The system is represented as a single operator evolution equation in a Hilbert space (the state space). The dynamics generator of this system is a non-selfadjoint operator with a compact resolvent. In the first paper, the asymptotic formulas for the eigenvalues of the dynamics generator have been derived. In the present paper, the following results have been shown. 1) The set of the generalized eigenvectors of the dynamics generator is complete and minimal in the state space. 2) The set of normalized generalized eigenvectors forms a Riesz basis, which is quadratically close to an orthonormal basis (a Bari basis). 3) The exact controllability problem has been solved via the spectral decomposition method. Show more

Keywords: Electro-mechanical energy harvester, non-selfadjoint matrix differential operator, generalized eigenvectors, Bari basis, exact and approximate controllability

DOI: 10.3233/ASY-171413

Citation: Asymptotic Analysis, vol. 102, no. 3-4, pp. 119-156, 2017

Authors: Lu, Yong | Zhang, Zhifei

Article Type: Research Article

Abstract: In this paper, we study the asymptotic behavior of nonlinear Klein–Gordon equations in the non-relativistic limit regime. By employing the techniques in geometric optics, we show that the Klein–Gordon equation can be approximated by nonlinear Schrödinger equations. In particular, we show error estimates which are of the same order as the initial error. Our result gives a mathematical verification for some numerical results obtained in [SIAM J. Numer. Anal. 52 (2014 ), 2488–2511] and [Numer. Math. 120 (2012 ), 189–229], and offers a rigorous justification for a technical assumption in the numerical studies [SIAM J. Numer. …Anal. 52 (2014 ), 2488–2511]. Show more

Keywords: Klein–Gordon equation, non-relativistic limit

DOI: 10.3233/ASY-171414

Citation: Asymptotic Analysis, vol. 102, no. 3-4, pp. 157-175, 2017

Authors: Possamaï, Dylan | Royer, Guillaume

Article Type: Research Article

Abstract: We study the utility indifference price of a European option in the context of small transaction costs. Considering the general setup allowing consumption and a general utility function at final time T , we obtain an asymptotic expansion of the utility indifference price as a function of the asymptotic expansions of the utility maximization problems with and without the European contingent claim. We use the tools developed in [SIAM Journal on Control and Optimization 51 (2013), 2893–2921] and [Communications in Partial Differential Equations 40 (2015), 2005–2046] based on homogenization and viscosity solutions to characterize these expansions. …Finally we study more precisely the example of exponential utilities, in particular recovering under weaker assumptions the results of [SIAM Journal on Financial Mathematics 3 (2012), 433–458]. Show more

Keywords: Transaction costs, homogenization, viscosity solutions, utility indifference pricing, asymptotic expansions

DOI: 10.3233/ASY-171415

Citation: Asymptotic Analysis, vol. 102, no. 3-4, pp. 177-226, 2017

Authors: Lopes, Cinthia Gomes | dos Santos, Renatha Batista | Novotny, Antonio André | Sokołowski, Jan

Article Type: Research Article

Abstract: Contact problems with given friction are considered for plane elasticity in the framework of shape-topological optimization. The asymptotic analysis of the second kind variational inequalities in plane elasticity is performed for the purposes of shape-topological optimization. To this end, the saddle point formulation for the associated Lagrangian is introduced for the variational inequality. The non-smooth term in the energy functional is replaced by pointwise constraints for the multipliers. The one term expansion of the strain energy with respect to the small parameter which governs the size of the singular perturbation of geometrical domain is obtained. The topological derivatives of energy …functional are derived in closed form adapted to the numerical methods of shape-topological optimization. In general, the topological derivative (TD) of the elastic energy is defined through a limit passage when the small parameter governing the size of the topological perturbation goes to zero. TD can be used as a steepest-descent direction in an optimization process like in any method based on the gradient of the cost functional. In this paper, we deal with the topological asymptotic analysis in the context of contact problems with given friction. Since the problem is nonlinear, the domain decomposition technique combined with the Steklov–Poincaré pseudo-differential boundary operator is used for asymptotic analysis purposes with respect to the small parameter associated with the size of the topological perturbation. As a fundamental result, the expansion of the strain energy coincides with the expansion of the Steklov–Poincaré operator on the boundary of the truncated domain, leading to the expression for TD. Finally, the obtained TD is applied in the context of topology optimization of mechanical structures under contact condition with given friction. Show more

Keywords: Topological sensitivity analysis, contact problems, domain decomposition technique, Steklov–Poincaré operator

DOI: 10.3233/ASY-171416

Citation: Asymptotic Analysis, vol. 102, no. 3-4, pp. 227-242, 2017

Article Type: Other

Citation: Asymptotic Analysis, vol. 102, no. 3-4, pp. 243-243, 2017