Modal analysis of elastic vibrations of incompressible materials using a pressure-stabilized finite element method
Introduction
Analyzing deformations in incompressible solid mechanics has become increasingly important in recent years due to its wide applications in industrial and research fields, and it is currently the subject of an active research, see [1], [2], [3], [4], [5], [6], [7], [8] and therein references. There are numerous methodologies devised to approximate the incompressible linear elasticity equations, including stabilized finite element methods (FEM) [9], [10], [11], [12], [13], discontinuous Galerkin methods [14], methods based on the least square approach [15], finite volume methods [1], collocation approaches [16], isogeometric approaches [17], [18], [19], and boundary element methods [20].
It is well known that the use of standard finite element approximations (as well as other approaches, see, e.g., [1], [21], [22] and the references therein) in elasticity problems is restricted due to the Poisson locking (dilatation), which is associated with the mathematical formulation being dependent on the Poisson ratio. In the limiting case where the Poisson ratio is equal to 0.5, the unknown of the problem (displacement) is divergence free, whose imposition in the formulation leads to the locking phenomenon (see, for instance [23], [24], [25], [26]). As opposed to the application in various structural models involving compressible materials, the incompressible media necessitate the incorporation of the pressure, or mean stress, into the model. In the standard Galerkin formulations, the displacement and pressure interpolations are required to satisfy the classical Babuška–Brezzi inf–sup condition [14], [22], [27], [28], [29]. These considerations apply to both the classical boundary value problem defining steady state incompressible elasticity (equivalently incompressible fluid flows) and the eigenproblem to be described in the sequel.
Aiming at avoiding the volumetric locking at the incompressible limit as well as circumventing the restrictions associated with the inf–sup condition, a great number of alternative stabilized finite element approaches have been proposed to solve the incompressible elasticity problems. An analysis of a mixed enhanced strain finite element method for the displacement–pressure formulation is presented in [30]. A mixed finite element method using primal and dual meshes is implemented in [29], where the standard space for the displacement is enriched with element-wise bubble functions. Another mixed formulation based on the dynamic variational multiscale approach is proposed in [31], where the momentum equation is complemented by a rate equation for the evaluation of the pressure. Stabilization strategies based on mesh-free polynomial projection methods are considered in [32], [33]. In [27], a pressure-curl stabilization approach is proposed in which the determination of the pressure stabilization parameter is based on stability concerns, and the curl stabilization parameter is determined on the account of dispersion.
In the case of transient problems, modal analysis constitutes an efficient alternative that is widely implemented to handle vibration problems of elastic materials (see, e.g., [2], [34], [35], [36]). In particular, let us cite the recent work [6], where a finite element method with discontinuous pressure basis functions is implemented to study the free vibrations of incompressible rectangular plates.
In this paper, our main objective is to present a modal analysis technique to simulate the linear elastic behavior of incompressible elastic solids where the incompressibility constraint is enforced by incorporating the pressure. The ultimate aim is to extend the robustness and effectiveness of the modal analysis in the mixed finite element framework recently proposed in [37]. In the system of transient elasticity equations consisting of a second order temporal derivative, a harmonic behavior of the displacement is assumed and each mode is considered to be of the form , where is the vector field of displacement amplitudes associated with the frequency . This is substituted into the equilibrium equations where the forcing terms are not considered in the case of free vibrations, yielding an eigenvalue problem (EVP) in which the eigenfunctions are the amplitudes , and the eigenvalues are the squares of the frequencies, . There exists a complete set of eigensolutions corresponding to positive eigenvalues as the elasticity operator is symmetric and positive definite, and hence, and the true solution can be expressed as a series of modes.
An eigenvalue problem has to be handled with a special precaution when it is approximated by using a stabilized finite element method. A residual based stabilization technique may lead to a quadratic EVP even if it is applied to approximate a linear EVP. We have proposed a FEM for the Stokes EVP that preserves the linearity of the continuous problem in [38]. The method is framed within the variational multiscale (VMS) concept, which assumes that the unknown can be split into a finite element component and a subgrid scale that needs to be modeled, and it has been applied to solve stationary boundary value problems in incompressible linear elasticity models in [11], [12], [13], [28], for example. The key point is to consider that this subgrid scale is orthogonal, in the -sense, to the finite element component. After approximating it, the result is a problem for the finite element component of the displacement amplitude and the pressure which permits any spatial interpolation. This yields an EVP that is linear, and that can be solved using arbitrary interpolations for the displacement and the pressure. We also remark here the possibility of alternative directions in approximating EVPs. An example is given in a recent study [8] in which two-field and three-field finite element least squares formulations are presented for EVPs associated with linear elasticity.
This paper is organized as follows. In Section 2 we describe the problem to be solved at the continuous level, both the original elastodynamic equations and the modal analysis, and considering both the differential and the weak form of the equations to be solved. The stabilized finite element approximation we propose is described in Section 3. Section 4 is concerned with the discrete modal analysis description, from which an approximate time integration scheme is presented and analyzed in Section 5. Numerical results are presented in Section 6, and finally conclusions are drawn in Section 7.
Section snippets
Statement of the problem
We consider the problem of modeling the vibrations of an incompressible linearly elastic body, assuming in particular infinitesimal strains. This initial and boundary value problem is considered to be defined on an open and bounded polyhedral domain , with boundary , , and time . It consists of finding a displacement field and a pressure field such that
Galerkin finite element approximation
Let us consider a finite element partition of the domain , with size . The collection of all edges of this partition is denoted as . From we may construct finite element spaces and to approximate and , respectively. We will restrict in the following to conforming approximations. Likewise, we shall need the space , constructed as but without prescribing the Dirichlet boundary conditions.
Let be the number of nodes to interpolate the
Discrete modal analysis
In the previous section we have presented the stabilized finite element formulation we propose to solve the EVP arising in the modal analysis of elastodynamics. Now we derive the expression of the solution of the discrete elastodynamic Problem (36)–(39) in terms of the eigenvalues and eigenfunctions obtained from Problem (40)–(41). The expression to be obtained is exact; we defer to the following section the truncation of this solution and the analysis of the underlying error.
We will prove that
Truncated solution and error analysis
The term modal analysis often refers to the approximate method in which only a few modes of expansion (42) are kept, that is to say, the solution is approximated as with (in the applications, ). The concern now is to bound the error associated to this approximation, and in particular to measure the difference . This will be done with norms associated to matrices.
Let be a symmetric
Numerical results
We present our numerical results on incompressible linear elasticity equations that confirm the theoretical analysis of the formulation proposed in this work. We focus ourselves on two different setups of plane stress problems, namely a rectangular cantilever beam and the well known Cook’s membrane problem are considered. We have chosen these examples because , for the cantilever beam and , for Cook’s membrane.
In the numerical experiments, we use quadratic type triangular
Conclusion
In this paper we have analyzed the modal analysis technique applied to elastic vibrations of incompressible materials. Incompressibility requires the introduction of pressure as a variable, so we have considered the displacement–pressure formulation. For the spatial approximation of the problem we have adopted a finite element method and, instead of adhering to the inf–sup condition between displacements and pressure to guarantee stability, we have presented a stabilized finite element
CRediT authorship contribution statement
Ramon Codina: Conceptualization, Methodology, Writing – review & editing. Önder Türk: Conceptualization, Software, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
R. Codina gratefully acknowledges the support received through the ICREA Academia Research Program of the Catalan Government, Spain. CIMNE is a recipient of a “Severo Ochoa Programme for Centres of Excellence in R&D” (grant CEX2018-000797-S) by the Spanish Government .
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