Eigenfrequency Problem Research Articles

A new adaptive penalization scheme for topology optimization

In the product development process topology optimization can successfully be applied in the early product design stages. In this work a new penalization scheme for the SIMP-method (Solid Isotropic Material with Penalization) is presented. One advantage of the presented method is a linear density-stiffness relationship, which has advantages for self-weight or eigenfrequency problems. The optimization problem is solved through derived optimality criteria (OC), which is also introduced in this paper. The derived OC uses no sensitivities of the optimization function but is equivalent to the classical OC in the case of no penalization. In the case with penalization we present an objective function for the penalty factor. By maximizing the objective function in each iteration, we get an adaptive penalty factor. Numerical experiments show, that the adaptive penalty factor leads to better optimization results then a fixed penalty factor. The presented topology optimization method is implemented in the commercial FEM package ANSYS under the name TopoAD. One useful extension of TopoAD is the concept of “meta elements”, which is also presented in this paper.

Torsional shear oscillations in the neutron star crust driven by the restoring force of elastic stresses

We present several exact solutions of the eigenfrequency problem for torsional shear vibrations in homogeneous and non-homogeneous models of the neutron star crust governed by the canonical equation of solid mechanics with a restoring force of Hookean elasticity. Particular attention is given to the regime of large-lengthscale nodeless axisymmetric differentially rotational oscillations, which are treated in spherical polar coordinates, reflecting the real geometry of the neutron star crust. We highlight the distinction between analytic solutions, computed as a function of the multipole degree of nodeless torsional oscillations and fractional depth of the crust, caused by different boundary conditions imposed on the toroidal field of material displacements. The relevance of the considered models to quasiperiodic oscillations, recently detected during the flare of SGR 1806-20 and SGR 1900+14, is discussed.

A New Boundary Element Analysis of 3-D Acoustic Fields Avoiding the Fictitious Eigenfrequency Problem

This paper is concerned with a new approach for avoiding the fictitious eigenfrequency problem in boundary element analysis of three-dimensional acoustic problems governed by Helmholtz equation. It is well known that, in solving without any care an external acoustic problem with internal sub-domains by means of the boundary integral equation, its numerical solution is violated at so-called fictitious eigenfrequencies corresponding to the internal sub-domains. The present paper proposes a new boundary element analysis to circumvent such a fictitious eigenfrequency problem by using dual boundary integral equation for nodal points on the boundary. One equation is the combined integral equation proposed by Burton-Miller. The other equation is the normal derivative boundary integral equation multiplied by the same coupling parameter as in the Burton-Miller expression. The quadrilateral element is employed in this study, and the Burton-Miller combined boundary integral equation is used at the middle nodes of element, while only the normal derivative boundary integral equation multiplied by the same coupling parameter is applied to the vertex nodes of element, and vice versa. The proposed approach is implemented, and its validity and effectiveness are demonstrated through numerical computation of the typical problems.

A New Design Sensitivity Analysis of Acoustic Problems Based on Bem Avoiding Fictitious Eigenfrequency Issue

This paper presents a new design sensitivity analysis based on the boundary element method avoiding the fictitious eigenfrequency issue in acoustic problems. The direct differentiation method is applied to the derivation of sensitivity formulas. In solving an external acoustic problem with internal sub-domains by means of the boundary integral equation without any care, is numerical solution is violated at the so-called fictitious eigenfrequencies corresponding to the internal subdomains. The present paper proposes a new boundary element sensitivity analysis avoiding such a fictitious eigenfrequency problem. This is based on the dual boundary integral equation, proposed previously by the authors, and the integral expressions are differentiated directly with respect to the design parameters. One equation is the combined boundary integral equation proposed by Burton-Miller and the other is the normal derivative boundary integral equation multiplied by the same coupling parameter as in the Burton-Miller expression. The quadrilateral element is employed in this study. The Burton-Miller integral expression is used only at the middle nodes of element, while the normal derivative boundary integral equation multiplied by the same coupling parameter is applied to the vertex nodes of element, and vice versa. The effectiveness of the proposed approach is illustrated through some numerical examples for three-dimensional problems.

A Boundary Element Analysis for Avoiding the Fictitious Eigenfrequency Problem in Acoustic Field (2nd Report, Revised Version)

This paper is concerned with an approach for avoiding the fictitious eigenfrequency problem in boundary element analysis of two-dimensional acoustic problems governed by Helmholtz equation. It is well known that in solving without any care the external acoustic problem by means of the boundary integral equation, the solution is violated at the eigenfrequencies of the internal problem. The present paper shows the effectiveness to circumvent the fictitious eigenfrequency problem by using the combined integral equation, which is originally proposed by Burton-Miller for constant elements, when employing the quadratic boundary element. On the other hand, we also propose a new approach to avoid that problem, which is for the quadratic boundary element. In this new approach, the Burton-Miller integral equation is used at the end points of element, while the normal derivative boundary integral equation multiplied by some parameter is applied to the other node of element. The proposed approach is implemented, and its effectiveness is demonstrated through comparison of the numerical results obtained by the developed computer code with other solutions.

An optimality criterion for shape optimization in eigenfrequency problems

For a broad class of static problems an optimality criterion of constant energy density at the designed boundary is known. In the present paper we prove a similar criterion for eigenfrequency problems. This optimality criterion serves as the tool for more basic understanding and for idealized reference cases as well as the basis for recursive procedures. Eigenfrequencies for in-plane vibrations as well as for out-of-plane vibrations of plates are optimized. The focus is on simplicity and multiple eigenfrequencies are not considered.

A New Method for Avoiding the Fictitious Eigenfrequency Problem in Boundary Element Analysis of Acoustic Problems (Consideration on 2D Problems)

This paper is concerned with a new method for avoiding the fictitious eigenfrequency problem in boundary element analysis of two-dimensional acoustic problems govened by Helmholtz equation. It is well known that in solving the external acoustic problem by means of the boundary integral equation, the solution is violated by the fictitious eigenfrequencies of the problem, which correspond to those of the internal problem. The present paper proposes a new method to circumvent the fictitious eigenfrequency problem by using the dual boundary integral equation for nodal points on the boundary. The quadratic boundary element is employed in this study. The usual boundary integral equation is applied to the end points of boundary element connected to other elements, while the derivaive boundary integral equation is applied to the other internal point of element. The proposed approach is implemented, and its effectiveness is demonstrated through comparison of the numerical results obtained by the developed computer code with other solutions.

Eigenfrequency optimization of stiffened plate using Kriging estimation

This paper describes the optimization of the beam stiffeners attached to plates in an eigenfrequency problem. The solution space is estimated using the Kriging method. A finite element analysis is carried out to evaluate the objective function at the sample points used for the estimation. The gradient method is used as a local optimizer. The Kriging estimation incurs relatively low cost, and is easy to combine with the gradient method. In this paper, we solve eigenfrequency optimization problems for a fully supported plate to maximize the minimum eigenfrequency and the difference between the 1st- and 2nd-order eigenfrequency. An optimization problem for an L-shaped plate with an eigenfrequency constraint is also solved. Good solutions are obtained for each example, and all the optimizations for these problems can be done at a lower computational cost. The results highlight the effectiveness of the method to solve the eigenfrequency optimization problems for the stiffened plate.

Structural optimization using Kriging approximation

An optimization method using Kriging approximation is applied to a structural optimization problem. The method involves two main processes. The first is a space estimation process that uses the Kriging method, and the second is an optimization process. The use of the Kriging method makes it easier to perform the approximation optimization. As an example of the estimation performed as part of structural optimization, a response surface for layout optimization of beam reinforcement is estimated. To evaluate the applicability of the Kriging method, Kriging estimation is compared with neural network approximation. As a numerical example, the optimization of a stiffened cylinder for an eigenfrequency problem is illustrated. The obtained results clearly show the applicability of the method.

Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams

An analytical study for dynamic behavior of pultruded fiber-reinforced plastic (FRP) composite cantilever I-beams is presented. Based on a Vlasov-type linear hypothesis, dynamic beam mass and stiffness coefficients, which account for both cross-section geometry and material anisotropy of the beam, are obtained. The eigenfrequency problem is solved by a Ritz energy method, and both exact transcendental and polynomial shape functions satisfying the boundary conditions of cantilever beams are used to describe the modal shapes. Good agreement between the proposed analytical method and finite-element analysis is obtained. The effect of beam span length, fiber orientation, and fiber volume fraction on natural frequencies is investigated. The proposed analytical solution can be used to effectively predict the vibration behavior of FRP cantilever I-beams.

Symmetries and exact solution of the molecular vibration problem

Utilizing the icosahedral symmetry of the C60 molecule by amalgamating two methods, we solve the vibrational eigenfrequency problem for carbon-60 to obtain exact algebraic expressions for the characteristic polynomials of the vibration. The inter-atomic interaction consists of four force constants, two for bond-stretching and two for angle-oscillation.

The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems

The method of fundamental solutions (MFS) is applied to acoustic scattering and radiation for axisymmetric bodies and boundary conditions. The fundamental solution of the governing equation and its normal derivative, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals, which are evaluated using library software. The method is tested on several problems from the literature and the results compared with existing solutions. Numerical experiments demonstrate that the fictitious eigenfrequency problem which is encountered with the boundary element method is not present in the MFS.

The Development of Boundary Fit Voxel Element

In the analysis of 3 dimensional solid structure, the mesh generation is always the most time consuming and sometimes makes the analysis impossible. Here, the authors have developed the boundary fitted voxel element to reduce the cost of mesh generation in 3D solid analysis. The shape functions that satisfy the continuity between elements are proposed, and integration method to compute the stiffness matrix, mass matrix and load vector of complicated shape is proposed. The element are implemented to MSC/NASTRAN as user defined element, and the unified numbering method of element and nodes for voxel elements are proposed. Also, the postprocessing technique, which does not use the finite element mesh, is proposed, and implemented using VRML. The analysis examples of static and eigenfrequency problems are shown.

Exact solution of the vibration problem for the carbon-60 molecule

Utilizing the icosahedral symmetry of the C 60 molecule by amalgamating two methods, we solve the vibrational eigenfrequency problem for carbon-60 to obtain exact algebraic expressions for the characteristic polynomials of the vibration. The inter-atomic interaction consists of four force constants, two for bond-stretching and two for angle-oscillation.

Polynomial particular solutions based boundary element analysis of acoustic eigenfrequency problems

The boundary element formulation for acoustic eigenfrequency analysis based on treating the forcing function, in the governing differential equation, as an initially unknown distributed source within the domain is presented. The volume integral due to this distributed source is eliminated by approximating the internal source by global interpolation and polynomial function representations and finding particular solutions for the governing, inhomogeneous equations. The resulting non-symmetric system matrix is solved by using Arnoldi's algorithm, modified to take advantage of the special structure of the substructured boundary element method. The techniques described here are embedded in a computer program GPBEST, and the numerical results are obtained by using this computer code.

The approximate spectral projection method

In this paper we develop a general method for investigating the spectral asymptotics for various differential and pseudo-differential operators and their boundary value problems, and consider many of the problems posed when this method is applied to mathematical physics and mechanics. Among these problems are the Schrodinger operator with growing, decreasing and degenerating potential, the Dirac operator with decreasing potential, the ‘quasi-classical’ spectral asymptotics for Schrodinger and Dirac operators, the linearized Navier-Stokes equation, the Maxwell system, the system of reactor kinetics, the eigenfrequency problems of shell theory, and so on. The method allows us to compute the principal term of the spectral asymptotics (and, in the case of Douglis-Nirenberg elliptic operators, also their following terms) with the remainder estimate close to that for the sharp remainder.

A user's guide to fortran programs for Wigner and Racah coefficients of SU 3

An accurate numerical solver is developed in this paper for eigenproblems governed by the Helmholtz equation and formulated through the boundary element method. A contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem, so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations. In order to accelerate the solution procedure, the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared. Moreover, a wideband fast multipole method is implemented with a block IDR(s) solver to reduce the overall solution cost of the boundary element systems of equations with multiple right-hand sides. The Burton–Miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains. The actual effect of the Burton–Miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as α=i/k is confirmed through exterior sphere examples. Furthermore, the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method.

Vibration of rectangular plates with edge-beams

The present paper deals with vibration of thin, isotropic, rectangular plates stiffened along two parallel edges and simply supported along the other two. An exact solution is obtained by reformulating the initial eigenfrequency problem as one of determining the wave-number codirectionally with simply-supported edges, with three different forms of eigenfrequency wave-number relationships. Examples are worked out, indicating the effect of stiffener parameters.

In-Plane Vibrations of Thick Circular Rings With T or I Cross Sections

This paper deals with the problem of eigenfrequencies and eigenvectors for rings whose cross section may be decomposed in basic rectangular cross sections. The solution is derived from a solution of the in-plane eigenvalue problem for rectangular cross-section thick rings. A good agreement between theoretical results and experimental data is obtained.

Radiation from an Array of Transducers on a Finite Cylinder

It is difficult to obtain valid solutions for a large finite cylindrical array using the Helmholtz integral or source-density formulations because of the density of the eigenfrequencies of the interior of the body. Near these eigenfrequencies, it is difficult to obtain valid solutions. If a hole is drilled through the cylinder, the interior eigenfrequencies of the resulting finite annular cylinder can easily be found. If the hole is large enough so that the resulting wall thickness is less than a half wavelength, then there will be no eigenfrequencies below the sonar frequency and the eigenfrequency problem is avoided. An example is given showing that this hole has negligible effects on the radiated pattern. Beam patterns for an array of six bands on a finite annular cylinder three wavelengths high, 10 wavelengths in diameter, and 14 wavelength in wall thickness are compared with patterns for the same array on an infinite cylinder, for various depression angles. The finite and infinite cylinder patterns looked much alike, with diffraction oscillations added. The diffraction oscillations were seen on the main lobe only for depression angles deeper than 30°.