Mathematical Physics - Volume II - Numerical Methods

Chapter 2. Finite element method

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to the rate of change of variable ( u ). According to conservation laws, the total flux entering each subdomain is equal to zero, and the flux can enter the subdomain of a body in one of two ways: • As an internal source, defined by function f and • Through an external boundary. Depending on the nature of the problem, internal sources of flux include volumetric forces f (our case), heat sources, fluid sources, etc. In order for the constitutive equations to hold, the state variable u (displacement), needs to be a continuous function of z , with defined values at both ends of its one-dimensional domain. Now, let us explain the process of discretization into finite elements, using a one-dimensional body (bar), on a domain 0 < z < ℓ (along the length of the bar). The bar is made of two materials, one of which has a domain of 0 < z < z 1 and the other has z 1 < z < ℓ . Material modulus k has a simple discontinuity in point z 1 , but it remains continuous in all points left and right of it. Internal source distribution (function) f ( z ) also has a discontinuity at a point z 2 where a concentrated force with magnitude ¯ f , represented by a Dirac delta function is acting. In addition, let us assume there is a simple source distribution discontinuity at point z 3 . Thus, in order to preserve the continuity of the domain, we can divide it into four subdomains, and 5 nodes connecting them. In this way, the quantities within the subdomain are continuous, since their corresponding discontinuities are located within the nodes between them.

Let us now consider some of the basic terms of mathematical physics: (1) The flux must be maintained in every point within a body (bar).

Let ¯ z be an arbitrary point within a subdomain in the interval a < ¯ z < b , Fig. 2.1b. Boundary fluxes in this area are denoted by arrows in the figure, and this area also contains internal force sources f ( z ) . Thus, the conservation law is given by:

b Z a

P ( b ) − P ( a ) =

f ( z ) d z .

(2.2)

(2) The flux is continuous in all subdomain points. The limit value of expression (2.2) when a → ¯ z − and b → ¯ z + (limits a and b are contracted into point ¯ z ) is: lim b → ¯ z + P ( b ) − lim a → ¯ z − P ( a ) = 0 , (2.3) since f ( z ) is a limited function within the a < z < b interval. If the term jump function is introduced at point ¯ z ∥ P ( ¯ z ) ∥ = lim b → ¯ z + P ( b ) − lim a → ¯ z − P ( a ) (2.4) we can write the following: ∥ P ( ¯ z ) ∥ = 0 , (2.5) which indicates that, in the case there are no jumps, the flux is continuous in all point within a subdomain Ω i , i = 1 , 2 , 3 , 4.