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Theoretical Background
Published in Valery Rudnev, Don Loveless, Raymond L. Cook, Handbook of Induction Heating, 2017
Valery Rudnev, Don Loveless, Raymond L. Cook
The principle of minimum energy requires that the vector potential distribution corresponds to the minimum of the stored field energy per unit length. As a result of that assumption, it is necessary to solve the global set of simultaneous algebraic equations with respect to the unknown, for example, magnetic vector potential at each node. The formulation of the energy functional, its minimization to obtain a set of finite element equations, and the solution techniques (the solver) were created for both 2-D (Cartesian system) and axisymmetric cylindrical system.
Film rupture and partial wetting over flat surfaces with variable distributor width
Published in Science and Technology for the Built Environment, 2019
Niccolo Giannetti, Piyatida Trinuruk, Seiichi Yamaguchi, Kiyoshi Saito
The following mathematical construct of the problem deviates from the intuitive Newtonian vectorial formulation by relying on the more abstract Lagrangian analytical formulation of the variational principles. The variational approach is based on the speculative assumption that physical phenomena can be described by relying on the principles of economy (Cline 2017). Explicitly, application of the principle of minimum energy (conceptually represented by Equation 1) is based on the assumption that among the set of possible flow configurations for a given flow rate, the one with minimal energy is the most likely to occur. where E defines the total energy content (Equation 2), and ω is a generic geometric variable of the flow configuration. The principle is applied to the unit streamwise length δx of the flow configuration (Figure 2).
A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment
Published in Mechanics Based Design of Structures and Machines, 2019
Shahriar Dastjerdi, Yaghoub Tadi Beni
U is the strain potential energy and Ω is the generated energy by the external forces. According to the principle of minimum energy, a system is in equilibrium, in which its total potential energy variations are equal to zero. is the Winkler coefficient of the elastic foundation and is the Pasternak coefficient of the elastic foundation. Equation (10) can be expanded as follows: