Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates

Abstract

The port-Hamiltonian formulation is a powerful method for modeling and interconnecting systems of different natures. In this paper, the port-Hamiltonian formulation in tensorial form of a thick plate described by the Mindlin-Reissner model is presented. Boundary control and observation are taken into account. Thanks to tensorial calculus, it can be seen that the Mindlin plate model mimics the interconnection structure of its one-dimensional counterpart, i.e. the Timoshenko beam. The Partitioned Finite Element Method (PFEM 1 ) is then extended to both the vectorial and tensorial formulations in order to obtain a suitable, i.e. structure-preserving, finite-dimensional port-Hamiltonian system (PHs 2 ), which preserves the structure and properties of the original distributed parameter system. Mixed boundary conditions are finally handled by introducing some algebraic constraints. Numerical examples are finally presented to validate this approach. * andrea.brugnoli@isae.fr † daniel.alazard@isae.fr ‡ valerie.budinger@isae.fr § denis.matignon@isae.fr 1 PFEM stands for partitioned finite element method. 2 PHs stands for port-Hamiltonian systems.

A port-Hamiltonian approach to modeling the structural dynamics of complex systems

“…1. We note that (23) and (17) are, except for the choice of parameters, identical equations. Consequently, the port-Hamiltonian formulation will be identical to (21) when choosing p D := I p ∂ω/∂t and q D := ∂ω/∂z as the energy variables.…”

Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates

“…A classical result is the fact that the adjoint of the vector divergence is div * = −grad as stated in [27]. This may be generalized to the adjoint of the tensor divergence Div * = −Grad (see Theorem 4 of [26]). Considering that A is the composition of two different operators A = div • Div and that the adjoint of a composed operator is the adjoint of each operator in reverse order, i.e.…”

“…Consider the space of power variables B defined in (58) and the matrix differential operator J in (54). By theorem 2 in [26] the linear subspace D ⊂ B…”

“…For this element the field is computed using quintic polynomials whose degrees of freedom are the values of the function, its gradient and its Hessian at the vertex of each triangular element. To deal with mixed boundary conditions Lagrange multipliers have to be introduced (the reader can refer to [26], section 4.3 for an explanation). The multipliers are therefore discretized by using second degree Lagrange polynomials defined over the boundary H 1 r (P 2 , ∂Ω).…”

A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control

“…It has successfully been applied to 1-D and also n-D systems, linear and nonlinear systems, with uniform or space-varying coefficients; it enables to deal with scalar-valued fields, vector-valued fields and also tensor-valued fields. Wave equations are tackled in [4,25], Mindlin's or Kirchhoff's plate equations are considered in [2,3], the treatment of the shallow water equations together with a general presentation of the Partitioned Finite Element Method (PFEM) is to be found in [5]. However, only lossless open systems have been addressed up to now: thus, the present paper intends to enlarge the scope of PFEM to lossy open systems, based on dissipative closed systems.…”