International Journal for Numerical Methods in Engineering

Lee, Hae Sung; Hong, Yun Hwa; Park, Hyun Woo

doi: 10.1002/nme.2769pmid: N/A

This paper presents a new class of displacement reconstruction scheme using only acceleration measured from a structure. For a given set of acceleration data, the reconstruction problem is formulated as a boundary value problem in which the acceleration is approximated by the second‐order central finite difference of displacement. The displacement is reconstructed by minimizing the least‐squared errors between measured and approximated acceleration within a finite time interval. An overlapping time window is introduced to improve the accuracy of the reconstructed displacement. The displacement reconstruction problem becomes ill‐posed because the boundary conditions at both ends of each time window are not known a priori. Furthermore, random noise in measured acceleration causes physically inadmissible errors in the reconstructed displacement. A Tikhonov regularization scheme is adopted to alleviate the ill‐posedness. It is shown that the proposed method is equivalent to an FIR filter designed in the time domain. The fundamental characteristics of the proposed method are presented in the frequency domain using the transfer function and the accuracy function. The validity of the proposed method is demonstrated by a numerical example, a laboratory experiment and a field test. Copyright © 2009 John Wiley & Sons, Ltd.

Le, Phong B. H.; Mai‐Duy, Nam; Tran‐Cong, Thanh; Baker, Graham

doi: 10.1002/nme.2771pmid: N/A

In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23:1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.

Souza, Flavio V.; Allen, David H.

doi: 10.1002/nme.2773pmid: N/A

Multiscale computational techniques play a major role in solving problems related to viscoelastic composites due to the complexities inherent to these materials. In this paper, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local‐scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history‐dependent stress tensor at the global scale depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of viscoelastic solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once at each material point and each time step. The procedure is basically developed by relating the local‐scale displacement field to the global‐scale strain tensor and using first‐order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify the approach and demonstrate the model capabilities. Copyright © 2009 John Wiley & Sons, Ltd.

Groenwold, Albert A.; Etman, L. F. P.

doi: 10.1002/nme.2774pmid: N/A

In topology optimization, it is customary to use reciprocal‐like approximations, which result in monotonically decreasing approximate objective functions. In this paper, we demonstrate that efficient quadratic approximations for topology optimization can also be derived, if the approximate Hessian terms are chosen with care. To demonstrate this, we construct a dual SAO algorithm for topology optimization based on a strictly convex, diagonal quadratic approximation to the objective function. Although the approximation is purely quadratic, it does contain essential elements of reciprocal‐like approximations: for self‐adjoint problems, our approximation is identical to the quadratic or second‐order Taylor series approximation to the exponential approximation. We present both a single‐point and a two‐point variant of the new quadratic approximation. Copyright © 2009 John Wiley & Sons, Ltd.

Mikš, Antonín; Novák, Jiří; Novák, Pavel

doi: 10.1002/nme.2783pmid: N/A

The quadrature of general, highly oscillatory integrals is a relatively complicated computational problem that occurs in a wide range of practical applications, e.g. in physics, chemistry, and image analysis. It is often necessary to use a high number of nodal points with classical quadrature formulas in order to achieve a required accuracy of numerical integration of rapidly oscillating functions. However, an increase in integration points leads to a larger computational time. Our work describes and analyses a method for numerical integration of rapidly oscillating functions, which enables to obtain the required accuracy with a smaller number of nodal points than classical integration rules and with a relatively simple integration formula. The proposed method is generally suitable for integration of rapidly oscillating functions in various applications. The method is demonstrated in examples of calculation of the diffraction integral in optics, where the integrand is often a rapidly oscillatory function. The results can be readily adapted to similar problems from other fields. Copyright © 2009 John Wiley & Sons, Ltd.