• Smart Structures and Systems
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• v.30 no.3
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• pp.221-237
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• 2022

This paper is concerned with nonlinear modeling and efficient numerical simulation of electrostrictive materials and structures. Two types of such materials are considered: relaxor ferroelectric ceramics and electrostrictive polymers. For ceramics, a geometrically linear formulation is developed, whereas polymers are studied in a geometrically nonlinear regime. In the paper, we focus on constitutive modeling first. For the reversible constitutive response under consideration, we introduce the augmented Helmholtz free energy, which is composed of a purely elastic part, a dielectric part and an augmentation term. For the elastic part, we involve an additive decomposition of the strain tensor into an elastic strain and an electrostrictive eigenstrain, which depends on the polarization of the material. In the geometrically nonlinear case, a corresponding multiplicative decomposition of the deformation gradient tensor replaces the additive strain decomposition used in the geometrically linear formulation. For the dielectric part, we first introduce the internal energy, to which a Legendre transformation is applied to compute the free energy. The augmentation term accounts for the contribution from vacuum to the energy. In our formulation, the augmented free energy depends not only on the strain and the electric field, but also on the polarization and an internal polarization; the latter two are internal variables. With the constitutive framework established, a Finite Element implementation is briefly discussed. We use high-order elements for the discretization of the independent variables, which include also the internal variables and, in case the material is assumed incompressible, the hydrostatic pressure, which is introduced as a Lagrange multiplier. The elements are implemented in the open source code Netgen/NGSolve. Finally, example problems are solved for both, relaxor ferroelectric ceramics and electrostrictive polymers. We focus on thin plate-type structures to show the efficiency of the numerical scheme and its applicability to thin electrostrictive structures.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.4
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• pp.160-167
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• 2005

Among finite filed arithmetic operations, division/inverse is known as a basic operation for public-key cryptosystems over $GF(2^{m})$ and it is computed by performing the repetitive $AB^{2}$ multiplication. This paper presents a digit-serial-in-serial-out systolic architecture for performing the $AB^2$ operation in GF$(2^{m})$. To obtain L×L digit-serial-in-serial-out architecture, new $AB^{2}$ algorithm is proposed and partitioning, index transformation and merging the cell of the architecture, which is derived from the algorithm, are proposed. Based on the area-time product, when the digit-size of digit-serial architecture, L, is selected to be less than about m, the proposed digit-serial architecture is efficient than bit-parallel architecture, and L is selected to be less than about $(1/5)log_{2}(m+1)$, the proposed is efficient than bit-serial. In addition, the area-time product complexity of pipelined digit-serial $AB^{2}$ systolic architecture is approximately $10.9\%$ lower than that of nonpipelined one, when it is assumed that m=160 and L=8. Additionally, since the proposed architecture can be utilized for the basic architecture of crypto-processor and it is well suited to VLSI implementation because of its simplicity, regularity and pipelinability.

• Communications of Mathematical Education
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• v.37 no.1
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• pp.65-84
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• 2023

The method of Lagrange multipliers, one of the most fundamental algorithms for solving equality constrained optimization problems, has been widely used in basic mathematics for artificial intelligence (AI), linear algebra, optimization theory, and control theory. This method is an important tool that connects calculus and linear algebra. It is actively used in artificial intelligence algorithms including principal component analysis (PCA). Therefore, it is desired that instructors motivate students who first encounter this method in college calculus. In this paper, we provide an integrated perspective for instructors to teach the method of Lagrange multipliers effectively. First, we provide visualization materials and Python-based code, helping to understand the principle of this method. Second, we give a full explanation on the relation between Lagrange multiplier and eigenvalues of a matrix. Third, we give the proof of the first-order optimality condition, which is a fundamental of the method of Lagrange multipliers, and briefly introduce the generalized version of it in optimization. Finally, we give an example of PCA analysis on a real data. These materials can be utilized in class for teaching of the method of Lagrange multipliers.