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Bailey, Cecil D.

doi:10.1023/a:1013857032364

Cited by 8 publications

(1 citation statement)

References 27 publications

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“…He also showed that integration of the differential equations of dynamics is in relation with the integration of a first order partial differential equation (the one for the generating function). The above relation can receive any convenient physical interpretation [21] provided it is remembered that the phase space and that of graphical optics are not isomorphous [22]. Hamilton applied the mathematics developed by himself to the conical refraction of biaxial crystals.…”

Section: Graphic Optics When Diffraction Is a Small Effectmentioning

confidence: 99%

Summary

Veselý¹

2016

Springer Theses

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Louis de Broglie's celebrated hypothesis transfers a problem of representation of optics to the quantum theories. Let us suppose that the fact that originated the problem in optics is the following, "The images obtained by optical instruments are limited by diffraction. Is the diffraction phenomenon exclusively restricting of the information contained in the image or does it have some physical meaning by itself?" This considered as a problem of measurement of the radiative field does not pose the question of tracing well-defined processes in the sense of the light quantum concept. Despite this the application thereto of quantum formalism makes the essentially inferring statistical nature of measurement forecasting most important. We suggest avoiding the probabilistic conception of measurement by replacing the wave-corpuscle dualistic principle with the duality one. The latter is a basic feature of projective geometry. Referring to it we shall try to include the diffraction in a geometrical model of optics and interpret in that coherent schema the point -plane "wave" dual solution. We show what experimental substantiation the proposed duality has. Lastly, we again discuss the experimental basis of wave-corpuscle dualism.

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Section: Graphic Optics When Diffraction Is a Small Effectmentioning

confidence: 99%

Summary

Veselý¹

2016

Springer Theses

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An accurate and fast regularization approach to thermodynamic topology optimization

Jantos

Hackl

Junker

2018

Numerical Meth Engineering

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In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii. KEYWORDSmeshless Laplacian, parabolic PDE, regularization, thermodynamic topology optimization, unstructured meshes, variational material modeling INTRODUCTIONOver recent decades, topology optimization has become a major field of interest in mechanical engineering. Approaches have been developed for different optimization objectives, constraints relating to physically feasible or manufacturable designs, or models for multiphysics problems. The most native approach is considered to be topology optimization with compliance minimization under a structure volume constraint for linear elastic materials, see, eg, the work of Bendsøe and Sigmund, 1 in which the elastic strain energy is subject to minimization and is referred to as (structural) compliance. Numerous solution approaches were developed of which the most popular are overviewed and compared in the work of Sigmund and Maute. 2 Usually, the topology has to be found within a given design space under mechanical boundary conditions, for which the mechanical problem is solved by a finite element approach. The design space contains material and void phases to identify the topology.This can be achieved either by tracing the phase borders or by introducing a density field that describes the material configuration in each point of the design space. The phase borders are subject to optimization in level-set approaches, 3,4 which Int J Numer Methods Eng. 2019;117:991-1017.wileyonlinelibrary.com/journal/nme

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An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

Junker

Balzani

2021

Continuum Mech. Thermodyn.

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An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

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Cited by 8 publications

References 27 publications

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An accurate and fast regularization approach to thermodynamic topology optimization

An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

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