Mathematical Physics - Volume II - Numerical Methods

2.1 Finite element application to solving of one-dimensional problems

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2.1.1 Variation formulation Classic representation of boundary problems using equations (2.18)-(2.20) requires solution regularity which is to strong for most real problems. In addition, equation (2.18) itself is not satisfied in discontinuity points ( z 1 , z 2 , z 3 ) , hence practical needs of numerical solving require nothing more than the use of weaker conditions for function u and its derivatives. Such approach to problem solving is referred to as weak or variation formulation, and is applicable to non-smooth data and solutions. Of course, if a smooth ("classical") solution to the problem exists, it also represents the solution of the weak problem. Hence, using this approach does not result in any loses, and allows the solving of most commonly encountered practical problems. Variation formulation is based on the need to find a function u, such that differential equation (2.18) and boundary conditions (2.20) are satisfied in the sense of pondered mean values: Z l 0 − ( ku ′ ) ′ v d z = Z l 0 f v d z Z l 0 − ( ku ′ ) ′ v d z = Z l 0 − ( ku ′ v ) ′ d z + Z l 0 ku ′ v ′ d z = (2.22) In expression (2.21), v is the weight function of z which is sufficient to ensure that integrals in this expression are of finite value. The set of all weight functions will be denoted as H . With this, we can provide a compact problem formulation as follows: Z l 0 [( ku ′ ) ′ + f ] v d z = 0 za v ∈ H , (2.23) u ( 0 ) = u 0 u ( l ) = u l . (2.24) If u and v functions are sufficiently smooth (at least in their subdomains), then partial integration provides the following: Z l 0 − ( ku ′ ) ′ v d z = Z l 0 ku ′ v ′ , d z − ku ′ v l 0 . (2.25) Having in mind the constitutive equation ( ?? ) we can replace ku ′ with P , and by further replacing (2.25) into (2.23), we obtain: Z l 0 ku ′ v ′ d z = Pv l 0 + Z l 0 f v d z . (2.26) Solution u belongs to a set ˜ H , called the test function class . This class’s functions must fulfil the essential boundary conditions, whereas natural boundary conditions occur automatically during the variation formulation of this problem. It should be noted that weight functions in points where essential BCs are defined equal zero. Taking into account that functions u and v appear in equation (2.26), they can be chose from the same class of allowable functions H = ˜ H = H 1 . (2.27) Since v can represent any given function from the allowable function set, we will consider a variant where u = v (= w ) . For simplicity’s sake, we will consider a case where k = const . Obviously, H 1 represents a class of functions w , where "1" denoted that all of its terms have first derivatives, whose squares are integrable at the interval of 0 < z < ℓ : Z l 0 ( w ′ ) 2 d z < ∞ . (2.28) = − ku ′ v l 0 + Z l 0 ku ′ v ′ d z .