Explicit Representation of SO(3) Rotation Tensor for Deformable Bodies

Computing the rotation tensor is vital in the analysis of deformable bodies. This paper describes an explicit expression for the SO(3) rotation tensor R of the deformation gradient F, and successfully establishes an intrinsic relation between the exponential mapping Q = exp A and the deformation F. As an application, Truesdell's simple shear deformation is revisited.

THE MAPPING OF ACTION VERBS IN THE TEACHING OF ENGLISH FOR FOOD PRODUCTION

This study is focused on the procedure of verbs’ translation in English (source language) into Indonesian language (target language), and how the mapping of action verb meanings in the procedural text. The research uses qualitative method, employing a cooking book recipe “Step by Step Cooking Balinese Delightful for Everyday” as its data source and its Indonesian translation. The theory used in this research is the theory of Vinay and Darbelnet (in Venuti, 2000) about translation procedures that include borrowing, calque, literal translation, transposition, modulation, equivalence, and adaptation. The theory of applying the natural semantic metalanguage approach (NSM) proposed by Wierzbicka (1996) is used to discuss the mapping of English action verbs. The theory is applied in order to explain how the Indonesian action verb meanings are mapped into English, with the exponential mapping technique. The description of the mapping meanings including the exponential mapping to the action verb of the Indonesian language has produced a new dimension. This new dimension turns out to be able to explore the meaning of the lexical item including the one that has even a subtle difference, therefore there is no more swirling of meaning. Keywords: translation procedure, action verb, mapping of meaning

Lamb Waves in Anisotropic Functionally Graded Plates: A Closed Form Dispersion Solution

ABSTRACTPropagation of harmonic Lamb waves in plates made of functionally graded materials (FGM) with transverse inhomogeneity is studied by combination of the Cauchy six-dimensional formalism and matrix exponential mapping. For arbitrary transverse inhomogeneity a closed form implicit solution for dispersion equation is derived and analyzed. Both the dispersion equation and the corresponding solution resemble ones obtained for stratified media. The dispersion equation and the corresponding solution are applicable to media with arbitrary elastic (monoclinic) anisotropy.

Coordinate Mappings for Rigid Body Motions

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

Group Preserving Correction Methods for Differential Algebraic Equations

The group preserving methods proposed by Liu [Int. J. Non-Linear Mech., 2001 and CMES-Comp. Model. Eng., 2006] for ordinary differential equations or differential algebraic equations (DAEs) adopted the Cayley transform or exponential mapping to formulate the Lie group from its Lie algebra. In this paper, we combine the Euler scheme with the group preserving methods to obtain the high accuracy group preserving techniques. We propose a group preserving correction scheme (GPCS) via exponential mapping and a modified group preserving correction scheme (MGPCS) by considering constraint. The two schemes provide single-step explicit time integrators for systems of DAEs. Some numerical examples are examined, showing that the GPCS and MGPCS work very well and have good computational efficiency and high accuracy.