China
Africa
Explore chapters and articles related to this topic
Wave, Diffusion and Potential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
where S denotes the surface of the domain of the problem. It is also referred to as the first boundary value problem in potential theory. For vibrations of a string, it gives a fixed end condition. For equilibrium problems of soap film, the Dirichlet problem is like a closed wire loop with prescribed deflection of the soap film. This problem has been studied by many well‐known mathematicians, including Poincare, Lyapunov, Gauss, Lord Kelvin, Weierstrass, Neumann, Wiener, Lebesgue, and Kellogg regarding its uniqueness and existence of the solution for different domains. In finite element formulation using calculus of variations, the Dirichlet boundary condition is normally referred to as the essential boundary condition (see Chapters 13 and 14). We will see in Section 9.7.8 that uniqueness of the solution ofthe Dirichlet problem can be guaranteed.
Applications of Partial Differential Equations
Published in Nita H. Shah, Mrudul Y. Jani, Partial Differential Equations, 2020
Here, x,y,z are Cartesian coordinates in space. The expression ∇2u is called the Laplacian of u. The theory of the solutions of the Laplace equation is called a potential theory. Solutions that have continuous second partial derivatives are known as harmonic functions.
Conformal Mapping
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
Because both the real and imaginary parts of F(z), namely, ϕ(x,y) and ψ(x,y), are harmonic functions, they satisfy Laplace’s equation and hence either one of these two could be used to find potential. Thus, the complex analytic function F(z) is known as complex potential. Laplace’s equation is one of the most important partial differential equations in engineering and physics. The theory of solutions of Laplace’s equation is known as potential theory. The concept of complex potential relates potential theory closely to complex analysis.
Measurement of wave forces on a modelled ice floe by plastic plate under bichromatic waves
Published in Ships and Offshore Structures, 2023
Longwei Huang, Wenyue Lu, Jianmin Yang, Qing Dong
The numerical simulation was performed using the DNV GL and Wadam software. The potential theory (Faltinsen 1993) was used to determine the wave force on the platform as follows: where and are the first- and second-order wave forces, respectively. These two force components can be calculated using Equations (5) and (6), respectively. where is the first-order transfer function; and are the second sum-frequency transfer function and second difference-frequency function, respectively; and , , , and are the wave amplitude, wave number, frequency, and random phase angle of the incident regular wave component, respectively.
Review of thermoelastic, thermal properties and creep analysis of functionally graded cylindrical shell
Published in Australian Journal of Mechanical Engineering, 2022
Ghassan F. Smaisim, Mostafa Omidi Bidgoli, Kheng Lim Goh, Hamed Bakhtiari
Omidi et al. (2019a) studied the influence of grading index on two-dimensional strain and stress distribution in functionally graded rotating cylinders resting on a friction bed and subjected to thermomechanical loads. In another work Omidi et al. (2019b) added an external torque and a thermal gradient on these cylinders and investigated their thermoelastic behaviour. Pakade, Dhakate, and Namdeo (2019) conducted an unsteady thermal analysis on a semi-infinite compact FGM cylinder. Parhizkar Yaghoobi and Ghannad (2020a, 2020b) used the first-order electric potential theory and perturbation technique to study the electro-elastic performance of functionally graded piezoelectric cylindrical shells with different thicknesses. Vaziri, Ghannad, and Anwar Bég (2019) suggested a precise solution for thermoelastic analysis of a thick-walled cylindrical shell under a non-uniform heat profile, using the first-order shear deformation theory. The material properties was radially distributed according to the power-law function and the method of separating variables and Bessel functions were utilised to solve the temperature field equations. The effect of the time on stress and displacements were calculated and compared with the results obtained from finite element simulations. In a recent study, Omidi Bidgoli et al. (2020) studied the unsteady stress and deformations in rotating functionally graded cylindrical shells made of Al-SiC and when subjected to mechanical and thermal loads.
Advances on simulation of wave-body interactions under consideration of the nonlinear free water surface
Published in Ship Technology Research, 2021
Daniel Ferreira González, Ulf Göttsche, Stefan Netzband, Moustafa Abdel-Maksoud
According to the comparison in Table 1, the results computed by panMARE match fairly with the HMEL method and the experiments. Relating to the case , , there is a significant difference of approximately for the second harmonic force between the panMARE results (|) and the experiments (). But a similar difference can also be stated between experiments and the higher order MEL approach (). This indicates that the overestimation of the horizontal force could be due to viscous effects or other limitations of potential theory.