The Variational Theory of Complex Rays applied to the shallow shell theory

doi:10.1016/j.compstruc.2015.05.021

Computers & Structures

Volume 158, 1 October 2015, Pages 98-107

https://doi.org/10.1016/j.compstruc.2015.05.021 Get rights and content

Highlights

•
VTCR is extended to shallow shell theory.
•
The uniqueness property of the shallow-shell-VTCR solution is demonstrated.
•
p-Refinement is the most performing.
•
VTCR outperforms FEM at mid-frequency.

Abstract

In the last few decades the interest of aerospace and automotive industries towards the study of the medium-frequency response of complex shell structure frames has grown. Recently some dedicated “wave” computational approaches have been developed. Among them, a Trefftz technique called Variational Theory of Complex Rays (VTCR) is catching on as an ad hoc method to deal with such vibration problems. This work presents the development of the VTCR in the shallow shell theory to increase its effectiveness and flexibility. First, general theory is given and two key properties of the solution demonstrated. After that, two numerical examples are deeply analyzed.

Introduction

In recent years, the interest of aerospace and automotive industries has been focused on efficient virtual testing of the vibration response. Shallow shell structures are widely used in these industrial contests due to their high resistance and light weight. The equilibrium equations of shallow shells are quite complex and in almost every real case an analytic solution cannot be obtained. Thus, an effective method to predict vibrational behavior in shallow shell structures is needed. The Modal Overlap Factor [1] defines three zones: low, mid and high frequency range. The low-frequency range has been extensively studied by the Finite Element Method (FEM) [2] and the Boundary Element Method (BEM) [3]. On the other side, the high-frequency range can be addressed by the Statistical Energy Analysis (SEA) [1]. This technique neglects almost entirely spatial quantities to focus on global energy. This effective approach is based on some key assumptions assured in the high-frequency range. The medium-frequency range is still an open question. On one hand the FEM and BEM are not indicated in this frequency domain since the phenomena variation length is very small if compared to characteristic dimensions of the structure. For this reason the required number of Degrees of Freedom (DoFs) explodes [4]. On the other hand the SEA is not suggested because the key assumptions of the theory might be unsatisfied [5]. Notwithstanding a lot of work has been done to extend such theories to the medium-frequency range [6], [7], [8]. There are also methods developed specifically for the medium-frequency range such as the partition of unity method [9], the ultra-weak variational method [10], the asymptotic scaled modal analysis [11], the energy operator eigenmodes [12], Galerkin method [13], the wave boundary element method [14] or the wave-based method [15], [16]. One of them is the Variational Theory of Complex Rays (VTCR). It approximates the vibrational problem solution with a sum of shape functions that identically satisfies the equilibrium equations while addressing the boundary conditions in weak form. This approach allows a priori independent approximations among subdomains. Thus, different (in number and type) shape functions can be chosen for each subdomain giving great flexibility to the method. It has already been applied to plate theory [17], to general shell theory [18], to transient dynamics [19], to 3D acoustic [20] and, on a wide frequency band [21], [22]. Nevertheless the shell version of the VTCR can still be improved. Yet the in-plane inertia was not taken into account in previous works, the weak variational formulation must be customized for the specific geometry, and the VTCR formulation does not address the general case of a boundary (or a corner) shared by multiple subdomains. Such problems are analyzed and solved in this work; the in-plane inertia assumption is relaxed and two propagative waves that lead in-plane stresses and displacements are introduced. The customization phase of the weak variational formulation is avoided using the shallow shell approximation providing effectiveness and flexibility to the method. Since in this theory the surface geometry is projected to the underlying area, the tuning phase of the weak variational formulation is no more needed. Finally, a more general version of the VTCR is presented.

The present work is structured as follows: first, the general theory is proposed providing some useful properties. After, two numerical examples are presented. The first one is an academic case where the analytic solution is known. Convergence tests are performed and performances are compared with a FEM reference. The second one is a complex structure frame.

Section snippets

Theory

In this Section the equilibrium and the boundary equations are examined using the standard shallow shell approximations. The theory is akin to the one provided in [23], [24]. After, some useful energy quantities are derived and the virtual work theorem is adapted for this specific case.

Shallow shell VTCR

In this Section the shallow shell version of the VTCR technique is illustrated. Since the VTCR is a Trefftz method, the solution is researched in the set of functions that satisfy the interior equilibrium equations. Boundary and corner residuals are addressed using Eq. (39). The weak variational problem definition is:

find the solution set $D_{si} = \{u_{si}, N_{si}, M_{si}\} \in D_{i}$ where $i \in [1, \dots, n]$ is the index related to the subdomain $Ω_{i}$ such that $\sum_{i = 1}^{n} \{\int_{\partial_{{\overline{v}}_{i}} Ω_{i}} δ b_{i}^{H} (v_{si} - {\overline{v}}_{i}) d s - \int_{\partial_{{\overline{b}}_{i}} Ω_{i}} δ v_{i}^{H} (b_{si} - {\overline{b}}_{i}) d s + \int_{\partial_{{\overline{w}}_{i}} Ω_{i}} δ q_{i}^{H} (w_{si} - {\overline{w}}_{i}) d s - \int_{\partial_{{\overline{q}}_{i}} Ω}\}$

Numerical examples

Two tests are presented in this Section. The first one studies a very simple vibrational problem where the analytic solution is known. The second is a complex frame structure where the VTCR solution is compared with a FEM one.

Conclusions

In order to increase its flexibility and effectiveness, VTCR was extended to shallow shell theory. The uniqueness property of the shallow-shell-VTCR solution was provided and two numerical examples were presented to validate the theory. The first one was an academic example where the analytic solution is known. Some convergence tests were run to analyze p- and h-refinements. p-Refinement proved to be the most effective in terms of total DoFs required as well as memory and time consumptions.

Acknowledgements

The authors gratefully acknowledge the “Centre National d’Études Spatiales (CNES)” and “Airbus Defense and Space”

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View full text

The Variational Theory of Complex Rays applied to the shallow shell theory

Highlights

Abstract

Introduction

Section snippets

Theory

Shallow shell VTCR

Numerical examples

Conclusions

Acknowledgements

J Sound Vib

Comput Methods Appl Mech Eng

Mech Syst Signal Process

Comput Methods Appl Mech Eng

Eng Anal Boundary Elements

Adv Eng Softw

J Sound Vib

Comput Struct

Comput Methods Appl Mech Eng

J Sound Vib

Statistical energy analysis of dynamical systems: theory and applications

The finite element method: linear static and dynamic finite element analysis

Boundary element method