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Eigenfrequency Optimization of Nonlinear Hyperelastic Structures using Low Density Element Removal

Dalklint, Anna LU (2018) In ISRN LUTFD2/TFHF-18/5230-SE(1-66) FHLM01 20182
Solid Mechanics
Abstract
Topology optimization with respect to eigenfrequencies is one example of an objective that has been studied extensively for small deformations. Few have however investigated this type of optimization for structures undergoing finite deformations. In this work, we thus expand the topology optimization formulations for eigenfrequency optimization to a nonlinear hyperelastic material model.

When working with topology optimization regarding eigenfrequencies one is commonly faced with two fundamental problems, namely multiple eigenfrequencies and spurious eigenmodes in low density regions. The first problem originates from the very definition of eigenvalues and corresponding eigenvectors. If an eigenvalue is considered multiple, the... (More)
Topology optimization with respect to eigenfrequencies is one example of an objective that has been studied extensively for small deformations. Few have however investigated this type of optimization for structures undergoing finite deformations. In this work, we thus expand the topology optimization formulations for eigenfrequency optimization to a nonlinear hyperelastic material model.

When working with topology optimization regarding eigenfrequencies one is commonly faced with two fundamental problems, namely multiple eigenfrequencies and spurious eigenmodes in low density regions. The first problem originates from the very definition of eigenvalues and corresponding eigenvectors. If an eigenvalue is considered multiple, the corresponding eigenvectors will loose their uniqueness i.e. any linear combination of the eigenvectors will also satisfy the original eigenvalue problem. The consequence of this phenomena arises in the sensitivity analysis and thus special care must be taken.

The second problem, i.e. spurious eigenmodes in low density regions, is automatically encountered when employing the SIMP penalization scheme to the stiffness of the finite elements to reduce intermediate density regions in the topology optimization discretization. Low material density elements will thus acquire a very small element stiffness, while keeping an unpenalized mass. Consequently spurious eigenmodes will form where the corresponding eigenfrequencies will take on very small values, which might affect the optimization procedure. There already exists means in the literature for dealing with these spurious eigenmodes, for example an interpolation scheme can be employed on the element mass matrix.

In this work we do however chose another way of eliminating theses spurious, localized eigenmodes. We employ an element removal method where low material density elements are entirely removed from the finite element discretization. Moreover, we implement a similar procedure as the one already existing to deal with multiple eigenvalues, which in addition incorporates the nonlinear responses. We perform topology optimization with displacement minimization as objective, together with constraints on a certain number of eigenfrequencies as well as the usual volume constraint on the available material.

The method of moving asymptotes is used to solve the topology optimization problem. Moreover, the Helmholtz PDE-filter is utilized to introduce a minimum feature size in the design and hence generate a well-posed topology optimization problem. From the numerical examples we conclude that the element removal method successfully can be used to eliminate spurious localized eigenmodes. Moreover we can conclude that by introducing constraints on the smallest eigenfrequencies, one can change the structural response. It can also be concluded that the magnitude of the load will influence the eigenvalues and hence it should be taken into account when analyzing prestrained nonlinear structures. (Less)
Popular Abstract
When trying to obtain the optimal structure in regards to performance and durability one cannot only consider the usual maximization of structural stiffness. If the structure is subjected to dynamical loads, for example vibrations, the distribution of the structural eigenfrequencies is also very critical for structural stability. Thus a new approach for designing nonlinear structures and/or materials with regard to their eigenfrequencies is presented.

Topology optimization can be defined as a mathematical method that tries to find the optimal distribution of material in order to maximize the performance of the structure. The sought after design is most often the stiffest structure, in other words one wants to find the distribution of... (More)
When trying to obtain the optimal structure in regards to performance and durability one cannot only consider the usual maximization of structural stiffness. If the structure is subjected to dynamical loads, for example vibrations, the distribution of the structural eigenfrequencies is also very critical for structural stability. Thus a new approach for designing nonlinear structures and/or materials with regard to their eigenfrequencies is presented.

Topology optimization can be defined as a mathematical method that tries to find the optimal distribution of material in order to maximize the performance of the structure. The sought after design is most often the stiffest structure, in other words one wants to find the distribution of material that copes the best with the applied load. However, it is not always the stiffest structure that performs the best, depending on what kind of loads and boundary conditions the structure is exposed to.

For example most structures in the real world are exposed to some kind of vibrations. If one is unlucky, the eigenfrequencies of the structure might reside in the spectrum of frequencies that the structure is exposed to. If this is the case the structure will start to vibrate which might induce annoying noise or even, in the long run, lead to the failure of the structure. Topology optimization with regard to eigenfrequencies is thus an important subject to investigate.

Others have previously proposed ways to, for example, maximize the smallest eigenfrequency of a structure using topology optimization. However, most of these approaches have only considered linear elastic structures, in other words structures consisting of materials acting linear elastic undergoing small deformations. Many structures in the real world are nonetheless exposed to such large load magnitudes that modeling them using the simplifications of linear elasticity would produce erroneous results.

In this work we thus extend the current formulations of eigenfrequency optimization to incorporate nonlinear responses. The proposed formulation succeeds in increasing the smallest eigenfrequency of the structure significantly. We also succeed in implementing a new way of dealing with problems linked to the void elements that come along with the topology optimization procedure. In this method these void elements are entirely removed from the finite element mesh. However, the full investigation of the impact of these void elements on the final solution has yet to be performed. Another thing that should be investigated further is whether different optimized designs really are obtained when introducing eigenfrequency optimization in comparison to the usual stiffness maximization. In the obtained results we succeed in increasing the smallest eigenfrequency, as mentioned before, however the impact on the final design is small. Thus a wider spectrum of initial conditions should be supplied to the proposed formulation, to hopefully end up in different design minimas. (Less)
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author
Dalklint, Anna LU
supervisor
organization
course
FHLM01 20182
year
type
H3 - Professional qualifications (4 Years - )
subject
keywords
Topology optimization, Eigenfrequency optimization, Nonlinear structures and materials, Element removal, Multiple eigenfrequencies
publication/series
ISRN LUTFD2/TFHF-18/5230-SE(1-66)
report number
TFHF-18/5230-SE
language
English
id
8964295
date added to LUP
2018-12-20 11:58:09
date last changed
2018-12-20 11:58:09
@misc{8964295,
  abstract     = {Topology optimization with respect to eigenfrequencies is one example of an objective that has been studied extensively for small deformations. Few have however investigated this type of optimization for structures undergoing finite deformations. In this work, we thus expand the topology optimization formulations for eigenfrequency optimization to a nonlinear hyperelastic material model. 

When working with topology optimization regarding eigenfrequencies one is commonly faced with two fundamental problems, namely multiple eigenfrequencies and spurious eigenmodes in low density regions. The first problem originates from the very definition of eigenvalues and corresponding eigenvectors. If an eigenvalue is considered multiple, the corresponding eigenvectors will loose their uniqueness i.e. any linear combination of the eigenvectors will also satisfy the original eigenvalue problem. The consequence of this phenomena arises in the sensitivity analysis and thus special care must be taken. 

The second problem, i.e. spurious eigenmodes in low density regions, is automatically encountered when employing the SIMP penalization scheme to the stiffness of the finite elements to reduce intermediate density regions in the topology optimization discretization. Low material density elements will thus acquire a very small element stiffness, while keeping an unpenalized mass. Consequently spurious eigenmodes will form where the corresponding eigenfrequencies will take on very small values, which might affect the optimization procedure. There already exists means in the literature for dealing with these spurious eigenmodes, for example an interpolation scheme can be employed on the element mass matrix.

In this work we do however chose another way of eliminating theses spurious, localized eigenmodes. We employ an element removal method where low material density elements are entirely removed from the finite element discretization. Moreover, we implement a similar procedure as the one already existing to deal with multiple eigenvalues, which in addition incorporates the nonlinear responses. We perform topology optimization with displacement minimization as objective, together with constraints on a certain number of eigenfrequencies as well as the usual volume constraint on the available material. 

The method of moving asymptotes is used to solve the topology optimization problem. Moreover, the Helmholtz PDE-filter is utilized to introduce a minimum feature size in the design and hence generate a well-posed topology optimization problem. From the numerical examples we conclude that the element removal method successfully can be used to eliminate spurious localized eigenmodes. Moreover we can conclude that by introducing constraints on the smallest eigenfrequencies, one can change the structural response. It can also be concluded that the magnitude of the load will influence the eigenvalues and hence it should be taken into account when analyzing prestrained nonlinear structures.},
  author       = {Dalklint, Anna},
  keyword      = {Topology optimization,Eigenfrequency optimization,Nonlinear structures and materials,Element removal,Multiple eigenfrequencies},
  language     = {eng},
  note         = {Student Paper},
  series       = {ISRN LUTFD2/TFHF-18/5230-SE(1-66)},
  title        = {Eigenfrequency Optimization of Nonlinear Hyperelastic Structures using Low Density Element Removal},
  year         = {2018},
}