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Filtering Frequencies in a Shift-and-invert Lanczos Algorithm for the Dynamic Analysis of Structures
Get PDFThe shift-and-invert Lanczos algorithm is a commonly used solution procedure to compute the eigenpairs of large, sparse eigenvalue problems that arise when approximating the elastic dynamic response of large structures under the influence of seismic forces. Not all eigenvectors are equally important to the response when the structure is exposed to a mass-dependent external force of the form , where is the mass matrix of the system and the rigid body vector. Structural engineers select eigenvectors , , such that their cumulative mass participation, measured as ,is above a target threshold . We show that when the starting vector for the unshifted Lanczos
algorithm is the spatial distribution vector , the Lanczos procedure can be used to provide an estimate of the cumulative mass participation. This allows us to identify intervals containing eigenvalues whose eigenvectors have a large contribution to the cumulative mass participation and filter out intervals containing eigenvalues whose eigenvectors have a negligible contribution. We use this information to devise a sequence of shifts for the shift-and-invert Lanczos algorithm as well as a stopping criterion for the iteration with shift so that the cumulative mass participation of the computed eigenvectors reaches the required level .
Numerical experiments on real engineering problems show that
our approach computes up to fewer eigenvectors and requires fewer shifts, on average, than the more general shifting strategy proposed by Ericsson and Ruhe (Math. Comp., 35 (1980)) together with its modification presented in Grimes et al. (SIAM J. Matrix Anal. and Appl. 40(4), 1994)
A block rational Krylov method for three-dimensional time-domain marine controlled-source electromagnetic modeling
No full textWe introduce a novel block rational Krylov method to accelerate three-dimensional time-domain marine controlled-source electromagnetic modeling with multiple sources. This method approximates the time-varying electric solutions explicitly in terms of matrix exponential functions. A main attraction is that no time stepping is required, while most of the computational costs are concentrated in constructing a rational Krylov basis. We optimize the shift parameters defining the rational Krylov space with a positive exponential weight function, thereby producing smaller approximation errors at later times and reducing iteration numbers. Furthermore, for multi-source modeling problems, we adopt block Krylov techniques to incorporate all source vectors in a single approximation space. The method is tested on two examples: a layered seafloor model and a 3D hydrocarbon reservoir model with seafloor bathymetry. The modeling results are found to agree very well with those from 1D semi-analytic solutions and finite-element time-domain solutions using a backward Euler scheme, respectively. Benchmarks of the block rational Krylov method demonstrate that it can be as up to 10 times faster than backward Euler. The block method also benefits from better memory efficiency, resulting in considerable speedup compared to non-block methods
Computing the Wave-Kernel Matrix Functions
Get PDFWe derive an algorithm for computing the wave-kernel functions and for an arbitrary square matrix , where . The algorithm is based on Pad\'{e} approximation and the use of double angle formulas. We show that the backward error of any approximation to can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for that is sharper than one previously obtained by Al-Mohy and Higham (\textit{SIAM J. Matrix Anal.\ Appl.}, 31(3):970--989, 2009). The amount of scaling and the degree of the Pade approximant are chosen to minimize the computational cost subject to achieving backward stability for in exact arithmetic. Numerical experiments show that the algorithm behaves in a forward stable manner in floating-point arithmetic and is superior in this respect to the general purpose Schur--Parlett algorithm applied to these functions
A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation
No full textThe theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be
derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium
Non-Abelian momentum polytopes for products of CP^2
No full textThis is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the point vortex systems familiar in 2-D. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 4-space. For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope for the product of 3 copies, and numerous transition polytopes, all of which are classified here. The different polytopes depend on the weights of the symplectic form on each copy of projective space. In the second paper we use reduction techniques to study the possible dynamics of interacting point vortices.
The results are also applied to determine the inequalities satisfied by the sum of up to three 3x3 Hermitian matrices with double eigenvalues
High Performance Software in Multidimensional Reduction Methods for Image Processing with Application to Ancient Manuscript
Get PDFMultispectral imaging is an important technique for
improving the readability of written or printed text where the letters have faded, either due to deliberate erasing or the ravages of time. Often the text can be read by illumination under a single wavelength of light, but in some cases the multispectral images need enhancement to improve the text clarity. There are many possible enhancement techniques: this paper compares an extended set of dimensionality reduction methods for image processing. We assess 15 dimensionality reduction methods applied to two different manuscripts. This assessment was performed subjectively, by asking the opinions of scholars who were experts in the languages used in the manuscripts, and also by using the Davies-Bouldin and Dunn indexes for evaluating the quality
of the resultant image clusters. We found that the Canonical Variates Analysis (CVA) method, implemented in Matlab was superior to all the other tested methods. However, the other approaches may be more suitable in specific circumstances, so we would still recommend that a variety are tried. For example, CVA is a supervised clustering technique and therefore it requires considerably more user time and effort than a non-supervised technique such as the Principle Component Analysis approach (PCA). If the results from PCA are adequate to allow a text to be read then the added effort required for CVA may not be justified. For the purposes of comparing the computational times and the image results, a CVA method is also implemented in the C
programming language and using the GNU (GNU’s Not
Unix) Scientific Library (GSL) and the OpenCV (OPEN
source Computer Vision) computer vision programming
library. Therefore high performance software was developed using the GNU GSL library, which drastically reduced the computational complexity and time for the CVA-GNU GSL method. For the CVA-Matlab technique, vectorization was used in order to reduce the respective computational times (i.e. matrix and vector operations instead of loop-based)
The relative compliance of energy-storing tendons may be due to the helical fibril arrangement of their fascicles
Get PDFA nonlinear elastic microstructural model is used to investigate the relationship between structure and function in energy-storing and positional tendons. The model is used to fit mechanical tension test data from the equine common digital extensor tendon (CDET) and superficial digital flexor tendon (SDFT), which are used as archetypes of positional and energy-storing tendons, respectively. The fibril crimp and fascicle helix angles of the two tendon types are used as fitting parameters in the mathematical model to predict their values. The outer fibril crimp angles were predicted to be 15.1° ± 2.3° in the CDET and 15.8° ± 4.1° in the SDFT, and the average crimp angles were predicted to be 10.0° ± 1.5° in the CDET and 10.5° ± 2.7° in the SDFT. The crimp angles were not found to be statistically significantly different between the two tendon types (p = 0.572). By contrast, the fascicle helix angles were predicted to be 7.9° ± 9.3° in the CDET and 29.1° ± 10.3° in the SDFT and were found to be statistically highly significantly different between the two tendon types (p < 0.001). This supports previous qualitative observations that helical substructures are more likely to be found in energy-storing tendons than in positional tendons and suggests that the relative compliance of energy-storing tendons may be directly caused by these helical substructures
Reduction and relative equilibria for the 2-body problem in spaces of constant curvature
Get PDFWe perform the reduction of the two-body problem in the two dimensional spaces of constant non-zero curvature and we use the reduced equations of motion to classify all relative equilibria (RE) of the problem and to study their stability. In the negative curvature case, the nonlinear stability of the stable RE is established by using the reduced Hamiltonian as a Lyapunov function. Surprisingly, in the positive curvature case this approach is not sufficient and the proof of nonlinear stability requires the calculation of Birkhoff normal forms and the application of stability methods coming from KAM theory. In both cases we summarize our results in the Energy-Momentum bifurcation diagram of the system
Effective Condition Number Bounds for Convex Regularization
Get PDFWe derive bounds relating the statistical dimension of linear images of convex cones to Renegar's condition number. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection to a lower dimensional space, and can still be effective if the linear maps are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality, interpreted as monotonicity property of moment functionals, and the kinematic formula from integral geometry. The main results are derived in the generalized setting of the biconic homogeneous feasibility problem