By P.A. Raviart
Книга из французской серии "Прикладная математика для магистрантов" дает основы математической теории метода конечных элементов.
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62) with suitable boundary conditions. Such a form may be used to solve problems ranging from temperature response in solids, seepage in porous media, magnetic effects in solids, and potential fluid flow. In the above, q is a flux and quite generally this can be written as q - q(r V r - - k(r V r 1 6 2 or, after linearization, dq = - k~162- k l d ( V r where kO = (~qi 0r and klj-- Oqi Or The source term Q(r also can introduce non-linearity. 63) -~6r162 6r q and is still valid if q and/or Q (and indeed the boundary conditions) are dependent on r or its derivatives.
In the forms given for problems in solid mechanics we will use interpolations which satisfy the C Orequirement; however, in the study of thin plates we will present some forms which violate the C 1 requirement. In addition to the continuity requirement it is necessary for C m functions to possess complete polynomials to order m + 1 to ensure that the derivatives up to order m + 1 can assume constant values. g. see reference 2 or 3). They remain equally valid for the study of non-linear problems - both for forms with material non-linearity as well as those with large deformations where kinematic conditions are non-linear.
4) has the free index i, thus to construct a weak form we multiply by an arbitrary vector with index i and integrate the result over the domain f2. Virtual work is a weak form in which the arbitrary function is a virtual displacement ~ui, accordingly using this function we obtain the form (~I'Ieq -- f ~ t~ui [pu i - o'ji,j -- b i ] d S2 : 0 Generally stress will depend on strains which are derivatives of displacements. Thus, the above form will require computation of second derivatives of displacement to form the integrands.