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complete three-dimensional numerical treatment of such problems is not practical, since the discrete problems are both very large and ill-conditioned Therefore, several classical assumptions were introduced long time ago in order to simplify the solution of plate problems There are two basic plate models, of which the historically older Kirchhoff (thin) plate model, based on the early works of Sophie Germain [24, 1191 and [96] in 1811, was completed and formalized by G Kirchhoff [71] in 1851 The thin plate assumptions were relaxed by E Reissner [97] in 1945 and in a slightly different way by R D Mindlin [81] in 1951 The Reissner-Mindlin plate model extends the field of application to shear-deformable thick plates, and therefore sometimes it is referred to as thick or shear-deformable plate model.

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The Reissner-Mindlin plate model is a second-order problem that naturally corresponds to the Timoshenko beam model, while the Kirchhoff model is a fourth-order problem that generalizes the Euler-Bernoulli beam model Although at the first glance the numerical treatment of the Reissner-Mindlin plate seems to be easier, it turns out that it conceals all basic difficulties plaguing the Kirchhoff model, and in reality its numerical solution is in some sense even more difficult On the other hand, the variational formulation of the fourth-order Kirchhoff model takes place in the space H2(R)which is much smaller than the space H1(R) In order to conform to H2(R),the approximations have to be once continuously differentiable (see Paragraph A43) Since the assembling procedures for the H2-conforming elements are nontrivial, nowadays it is popular to resort to mixed methods that lead to the standard HI-conforming elements (see, eg, [18,95] and [ 1241).

(3.7) and note that it can be expressed in matrix form as follows:

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Instead of reviewing existing results on mixed methods, we find it more useful to focus on the application of the less frequently encountered C'-elements in this text These elements are natural for the variational setting in the space H2(R)It was demonstrated in Paragraphs 637 and 643 that the key to a transparent element-by-element assembling procedure are the correct transformation relations for the weak forms and shape functions from the mesh element to the reference domain and vice versa These topics are addressed in Section 66 Prior to introducing the finite elements, in Paragraphs 651-654 we present the derivation of the thick and thin plate models, list various types of boundary conditions, construct the variational formulation and discuss the existence and uniqueness of the weak solution.

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It turns out that the approximation of a smooth boundary via a nonsmooth curve (a common technique for second-order problems) may change the physics of the fourth-order problem completely This phenomenon, known as the BabuSka's paradox of thin plates, is mentioned in Paragraph 655..

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