Ki-Ook Kim

Efficient higher-order shell theory for laminated composites

An efficient higher-order shell theory is obtained for symmetric laminated composites. The in-plane displacement fields are obtained by superimposing a globally cubic varying displacement field on a zig-zag linearly varying one. For an orthogonal curvilinear coordinate system, equilibrium equations and boundary conditions are derived using lines of curvature coordinates. Cylindrical shell equations are obtained from the general equilibrium equations. To evaluate the present shell modeling, the analytical solution for a cylindrical bending problem is obtained. The present shell theory gives deformation and stresses which are in good agreement with those of exact elasticity solutions.

A REVIEW OF MASS MATRICES FOR EIGENPROBLEMS

Abstract Various nonconsistent mass matrices have been presented to achieve more accurate natural frequencies in eigenproblems of the finite element analysis. The matrices are obtained as a linear combination of lumped and consistent mass matrices. For an improved accuracy, the consistent mass should be more weighted than the lumped mass. Instead of the mass combination, the interpolation functions can be combined to give nonconsistent mass matrices, which show the same tendency. To find a nonsingular lumped mass matrix for the bending vibration of beams, a translational inertia has been proposed for rotational degrees of freedom. The inertia effect is highly overestimated and hence lower natural frequencies are obtained. When combined with the consistent mass matrix, however, the modified lumped mass matrix gives a significant improvement for the natural frequencies of intermediate and higher modes. A simple corrective method was applied to get a better estimation of the natural frequencies through the use of the frequency dependent stiffness and mass matrices. The method shows high accuracy without complicated calculations.

Energy Method for Selection of Degrees of Freedom in Condensation

An analytical method is presented for the selection of degrees of freedom in condensation of eigenproblems. The method isbasedon theenergiesassociatedwiththedegrees offreedom in theeigenmodesofstructuralsystems. For theenergyestimationRitzvectorsarecalculated,usingthestiffnessandmassmatrices.Theenergiesaddedthrough themodesortheweighted rowsum oftheenergy distribution matrix can be used asan effective guideline on which degrees of freedom should be retained in the analysis. Another approach of sequential selection can be employed, in which a e nite number of new degrees of freedom with the largest energy are taken in each mode and the e nal union becomes the analysis set. The energy of the selected degrees of freedom or the column sum of the matrix can beused to predict thesolution accuracy in each mode. The erroranalysisshowsthattheperturbation in eigenvalue is related with the energy of the degrees of freedom. The row and column sums indicate the completeness of the selected set and give a clue to how many degrees of freedom to be included. The energy criterion has shown that the conventional practice of choosing translational degrees of freedom is appropriate only for the lowest modes. Numerical investigations were performed to test the convergence criterion. The energy method proved to work well for typical example problems.

Convergence Acceleration of Iterative Modal Reduction Methods

An accelerated method is presented for the iterative condensation of eigenproblems. The present study was motivated by the improved reduction system and the succession-level approximate reduction (SAR). The reduction procedures are supplemented with the second-order approximation in the series expansion of the system transformation. The reduced equation of an equivalent system and the transformation matrix are updated in an iterative manner.In addition,systematicderivation andcomparisonoftheequationsinvolvedin variouscondensationshave been sought.Thematrixupdateincorporatesnotonlyinverseiteration butalsosubspacetransformation implicitly. Theseriesexpansion canbeconsideredasrepeatedupdatesofthetransformationmatrixthroughinverseiteration. The solution accuracy is sensitive to the selection of the degrees of freedom, for which sequential elimination or energy method may be preferable. When a poor selection causes a sudden failure of the updatemethod, the hybrid dynamic condensation can be used. The method of SAR is closely related to the hybrid dynamic condensation. Nomenclature [As] = [kss] i 1 [mss] in Eq. (10) [Bpp],[Bsp] = submatrices for inertia force in Eq. (68) [DG] = [MG] i 1 [KG] dynamic matrix in Eq. (13) [Ke], [Me] = equivalent structural matrices in Eq. (26) [KG],[MG] = reduced matrices in Guyan’ s condensation [Kr], [Mr] = structural matrices reduced through [Tr] [KX], [MX] = structural matrices reduced through [TX] [k], [ka b ] = stiffness matrix and submatrices [m], [ma b ] = mass matrix and submatrices p = number of primary degrees of freedom [Q pp] = modal matrix for generalized coordinates {R} = residual error in eigenproblem s = number of secondary degrees of freedom [T] = exact transformation matrix in Eq. (22) [Ti] = matrices in series of Eqs. (15) and (18) [Tp] = system transformation matrix [Tr] = approximate transformation matrix [TX] = improved transformation matrix in Eq. (40) [T0] = transformation in Guyan’ s condensation [T(k )] = frequency-dependent transformation in Eq. (3) [X p] = improved modal matrix [X pp],[Xsp] = submatrices of [X p] e = convergence tolerance [K p]

Direct Approach in Inverse Problems for Dynamic Systems

A direct and effective approach is presented for the inverse problem of dynamic structural systems, which is related to structural optimization, system identification, and damage detection. The structural modifications are sought for the characteristic changes assigned from the design goals or modal measurements. A finite element method is used for the system analysis and inverse problem. Mathematical programming techniques are applied for the minimization of the deviation of the finite element model from the desired inverse system, along with an objective function of least structural change. The modal method is based on the perturbation equations of a set of selected degrees of freedom and the energy equation associated with the frequency change. The mode shape change is expressed as the sum of the baseline mode shape and complementary vector, which plays a very important role in the search for the inverse solution. The linear perturbation equation is employed to get an initial approximation, which can be improved through iterations with the nonlinear perturbation equation. The proposed method does not involve the system reduction. Therefore, it is believed to be superior to conventional methods, which suffer from the residual error due to transformation of the eigenproblem.

System condensations for inverse problems of linear dynamic structures

A modal method combined with system condensation is presented for the assessment of selection optimality and solution accuracy in inverse problems of structural optimization and damage detection. Finite element procedures are applied to seek the structural modifications for the characteristic changes assigned from design goals or dynamic measurements. The solution convergence is related to the selection of degrees of freedom and the method of system transformation. The application of the dynamic stiffness matrix yields a frequency-dependent transformation matrix, which can be expanded into an infinite series to obtain lower-order approximations. The modal matrix may be used to project the measured data onto the mode shapes, in which case much emphasis is laid on the linear independence of the selected degrees of freedom and the condition number of the transformation matrix. The baseline structure is used to obtain an initial perturbation, which can be improved through repeated updates of the transformation. The proposed methods give excellent solutions for frequency optimization. In damage detection, however, moderate deviations from the correct structural changes are attributable to system reduction. The dynamic stiffness matrix seems recommendable over the modal matrix projection for the system transformation.

Study of Effects of Measurement Errors in Damage Detection

A modal method is presented for the investigation of the effects of measurement errors in damage detection for dynamic structural systems. The structural modifications to the baseline system result in the response changes of the perturbed structure, which are measured to determine a unique system in the inverse problem of damage detection. If the numerical modal data are exact, mathematical programming techniques can be applied to obtain the accurate structural changes. In practice, however, the associated measurement errors are unavoidable, to some extent, and cause significant deviations from the correct perturbed system because of the intrinsic instability of eigenvalue problem. Hence, a self-equilibrating inverse system is allowed to drift in the close neighborhood of the measured data. A numerical example shows that iterative procedures can be used to search for the damaged structural elements. A small set of selected degrees of freedom is employed for practical applicability and computational efficiency.

Self-equilibration of inverse system in damage detection for dynamic structures

In this study, a modal method coupled with mathematical programming is presented for the assessment of error propagation in damage detection for linear dynamic structural systems. Finite element method is applied to inverse problems in which the structural modifications to the baseline system are sought for the characteristic changes of the perturbed structure. Unlike structural optimization, system identification and damage detection require a unique inverse system. When the exact modal responses are available, accurate structural changes can be obtained through iterative procedures with nonlinear perturbation equations. In practice, however, the modal data may contain numerical errors and consequently yield incorrect inverse solutions that are physically infeasible or quite different from the real perturbed system. Due to the intrinsic instability of eigenproblem, minor errors tend to cause significant element deviations. Hence, mathematical programming techniques are applied to search for a self-equilibrating inverse solution in the close vicinity of the measured data. The proposed approach may provide a breakthrough in the effort to alleviate the numerical difficulty of error propagation in the modal method for the inverse problem of damage detection. The linear perturbation equation is used to obtain an initial approximation, which can be improved through nonlinear iterations.

Advancements of Aerospace Computational Structure Technology in Korea

Over the past decades, various research groups in Korea have made considerable research efforts in the field of aerospace computational structure technology. Nowadays, the main research efforts include high performance computing, pre/post-processor, novel computational methodologies, and application of computational structure technology to real design and development of aircraft and satellite. In this paper, some of the representative current research activities in Korea are presented.

Structural System Identification Using Iterative Improved Reduced System

In the inverse perturbation method, enormous computational resource was required to obtain reliable results, because all the unspecified DOFs were considered as unknown variables. Thus, in the present study, an iterative reduced system method is used to condense the unspecified DOFs to improve the computational efficiency as well as the solution accuracy. All the conventional reduction methods include transformation errors in the transformation matrix between the unspecified DOFs and the specified DOFs. Thus it is hard to obtain reliable and accurate solution of inverse perturbation problems by reduction methods due to the error included in the transformation matrix. This numerical trouble is resolved in the present study by adopting iterative improved reduced system (IIRS) as well as by updating the transformation matrix at every step in the inverse perturbation method. In this reduction method, system accuracy is related to the selection of the primary DOFs. The two-level condensation method (TLCS) is employed to select the proper primary DOFs for increasing accuracy and reducing iteration time. Numerical results of the present iterative inverse perturbation method (IIPM) are presented for the verification of the proposed reduced method in the inverse problem.