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Parametric Mesh Generation for Optimization
Published in S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole, Finite Elements-based Optimization, 2019
S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole
A Dirichlet boundary condition means the potential along the given boundary is fixed and a Neumann boundary condition means the derivative of the potential along the given boundary is fixed, and usually zero. Dirichlet boundary conditions can be implemented by keeping the potential of all the points on the given boundary to be fixed at their given value. The user can select any segment and define the potential of that segment. If the potential of the segment is set, then all the points which will be added onto that line will get this potential automatically. The boundaries that do not implement Dirichlet conditions will automatically act as Neumann boundaries during the FEA process. This is because it is natural to the finite element formulation (Hoole, 1988). Therefore, no special provisions are needed to define Neumann boundaries.
Computational Fluid Dynamics and Multiphysics
Published in Mariano Martín Martín, Introduction to Software for Chemical Engineers, 2019
Bostjan Hari, Borja Hernández, Mariano Martín
Every partial differential equation needs an initial value or guess for numerical solver to start computing the equations. On the other hand, boundary conditions are specific for each conservation equation, described in Section 8.2. The variable in the continuity equation and momentum equations is the velocity vector, the variable in the energy equation is the temperature vector and the variable in the species equation is the concentration vector. Therefore, appropriate velocity, temperature and concentration values, that represent real-world values, need to be prescribed on each computational boundary, such as inlet, outlet or wall at time zero. The prescribed values on boundaries are called boundary conditions. Each boundary condition needs to be prescribed on a node or line for two-dimensional system or on a plane for three-dimensional system. In general, there are several types of boundary conditions where the Dirichlet and Neumann are the most widely used in CFD and multiphysics applications. The Dirichlet boundary condition specifies the value on a specific boundary, such as velocity, temperature or concentration. On the contrary, the Neumann boundary condition specifies the derivative on a specific boundary, such as heat flux or diffusion flux. Once the appropriate boundary conditions are prescribed to all boundaries on the two-dimensional or three-dimensional model, the set of the conservation equations is closed and the computational model can be executed.
Boundary Conditions
Published in Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin, Finite Analytic Method in Flows and Heat Transfer, 2020
Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin
Orszag and Israeli [142] have pointed out that either the normal or the tangential component of the momentum equations is permissible as a boundary condition for the pressure Poisson equation. The former leads to a Neumann boundary condition and the latter to a Dirichlet boundary condition. Moin and Kim [133] found that the Neumann and Dirichlet boundary conditions generally may not provide the same solutions, contradicting the study of Orszag and Israeli. Gresho [77] gave a thorough review of pressure boundary conditions for the incompressible Navier-Stokes equations. He concluded that for internal flows with a Dirichlet velocity boundary condition, only a Neumann boundary condition is always appropriate for the pressure Poisson equation.
Full compressible Navier-Stokes equations with the Robin boundary condition on temperature
Published in Applicable Analysis, 2023
Let us convert the problem (1)–(4a) back to the Euler coordinates : The Robin boundary condition is a weighted combination of the Dirichlet boundary condition and the Neumann boundary condition. If Ω is the domain and denotes its boundary, is the exterior unit vector normal to . Then, the Robin boundary condition is for some constants a, b and a given function f defined on . In the present paper, , the Robin boundary condition (6) becomes Then, the condition (4a) includes the condition (7) with . The condition (4b) is equivalent to (7) with and .
Identifying unknown source in degenerate parabolic equation from final observation
Published in Inverse Problems in Science and Engineering, 2021
Thus, in the case of , if , then u satisfies the Dirichlet boundary condition and in the case , every satisfies the Neumann boundary condition and the Dirichlet boundary condition . Moreover, the operator A is the infinitesimal generator of a strongly continuous semigroup on . Consequently, we have the following well-posedness result (see, e.g. Futouhi and Salimi [9, Theorem 14]).
A multidomain multigrid pseudospectral method for incompressible flows
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Wenqian Chen, Yaping Ju, Chuhua Zhang
The Dirichlet or Neumann boundary condition is employed for velocity and temperature. Combining with the boundary condition for pressure (5), we denote the boundary conditions for unknowns by a general formulation: where, a, b, h are undetermined parameters related to a specific problem. a = 0 and b ≠ 0 denotes the Dirichlet boundary condition, a ≠ 0 and b = 0 Neumann boundary condition, and a ≠ 0 and b ≠ 0 Robin boundary condition. For periodic boundary condition, the physical boundary can be transformed to an artificial internal interface that can be treated with the multidomain method stated in next section.