By Michael Schäfer
This e-book is an creation to trendy numerical equipment in engineering. It covers purposes in fluid mechanics, structural mechanics, and warmth move because the such a lot proper fields for engineering disciplines corresponding to computational engineering, medical computing, mechanical engineering in addition to chemical and civil engineering. The content material covers all facets within the interdisciplinary box that are crucial for an ''up-to-date'' engineer.
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Additional info for Computational Engineering — Introduction to Numerical Methods
For this, additional diﬀerentiability properties of the displacements are necessary, which, however, we will not elaborate here. Another special case of linear elasticity theory, which can be described by a one-dimensional equation, is beam bending (see Fig. 9). We will focus here on the shear-rigid beam (or Bernoulli beam). This approximation is based on the assumption that during the bending along one main direction, plane crosssections remain plane and normals to the neutral axis (x1 -axis in Fig.
The most relevant assumptions for practical applications are the incompressibility and the inviscidity, which, therefore, will be addressed in detail in the following. 1 Incompressible Flows In many applications the ﬂuid can be considered as approximatively incompressible. , ∂vi /∂xi = 0 (see Sect. 1), and one speaks of an 44 2 Modeling of Continuum Mechanical Problems incompressible ﬂow. For a criterion for the validity of this assumption the Mach number Ma = v¯ a is taken into account, where v¯ is a characteristic ﬂow velocity of the problem and a is the speed of sound in the corresponding ﬂuid (at the corresponding temperature).
30) Possible boundary conditions for linear elasticity problems are: − Prescribed displacements: ui = ubi on Γ1 , − Prescribed stresses: Tij nj = tbi on Γ2 . , Γ1 ∩ Γ2 = ∅ and Γ1 ∪ Γ2 = Γ . 30) as a system of partial differential equations, there are other equivalent formulations for linear elasticity problems. We will give here two other ones that are important in connection with diﬀerent numerical methods. We restrict ourselves to the steady case. 30) with a test function ϕ = ϕi ei , which vanishes at the boundary part Γ1 , and integration over the problem domain Ω yields: (λ+μ) ∂ 2 uj ∂ 2 ui ϕi dΩ + +μ ∂xi ∂xj ∂xj ∂xj Ω ρfi ϕi dΩ = 0 .
Computational Engineering — Introduction to Numerical Methods by Michael Schäfer