A DDA Based Complete And High Order Polynomial Displacement Approximation Method In Elastic Mechanics And Its Cases Verification

According to Weierstrass theorem for polynomial approximation and based on elastic mechanics derivationwhere the complete and high order polynomial displacement function used in DDA is employed, a completeand high order polynomial function approximation method is presented in this thesis. This method assumesthat the displacement function of a block unit is a complete and high order polynomial function inthree-dimensional space, the unknown of the simultaneous equations is the coefficient column matrix in thecomplete high order polynomial function, and the equilibrium equations are established according to thedisplacement variation method in elastic mechanics. In addition, in the process of establishing globalequilibrium equations, the traditional numerical integration which tries to approximate the analytical solutionis substituted by simplex integration which is precisely the analytical solution. Therefore, the essence of highorder DDA method, regarding to this kind of problems in elastic mechanics, is actually addressing theproblems concerning with the complete and high order polynomial displacement approximation which tries toapproximate a certain unknown continuous displacement function under the same restriction of a series ofgoverning equations. According to the approximation theorem concerned with polynomial function which isproposed by the Mathematician, Weierstrass, inference could be made that, considering any problems aboutelastic mechanics, the calculating result would approximate the analytical solution monotonously along withthe increase of the displacement function order in DDA method. This thesis extends the original program,introduces the diagram interpretation for simplex integration, studies the displacement of simple structures andcomplex structures with connecting surface and compares the results with elastic analytical solutions andFinite Element Method. The research results mainly contain:(1) It introduces the mathematical foundation of complete and high order polynomial displacementapproximation, or the application of Weierstrass Theorem in this method. Concerning whether it should beapproximated by a higher order complete polynomial function while the objective function is a polynomial,this thesis proposes that it could effectively improve the accuracy of the results if higher order completepolynomial function is hired and the reasons are given in detail.(2) It extends the order of the complete polynomial in the original program from five to six, or the numberof the terms in displacement function increases from56to84, which approximate the objective displacementfunction more accurately.(3) The submatrix of uniform distributed loading is added in the original program, whose submatrices ofloading originally contain the submatrix of initial loading, the submatrix of point loading and the submatrix ofbody loading.(4) In this thesis, the diagram interpretation for simplex integration in three-dimensional space is given indetail based on the computing requirement in engineering, while the diagram interpretation in n-dimensionalspace is easy to be attained according that. It introduces the meaning of diagram interpretation for simplexintegration in detail, or it describes every product term and computing condition in the algebraic sum of theproduct in the simplex integration formula, which makes the complex integration procedure in the formulaeasier to be understood.(5) It analyzes the mechanical behavior of a single block with the written program. The cantilever beamand simply supported beam both with uniform distributed loading and a boundary fixed elastic plate with pointloading are analyzed; the displacement calculated by high order DDA method is compared with elasticanalytical solution; modification proposal for analytical solution of cantilever beam and the reasons for that is given in detail. All these cases verify the convergence and accuracy of this method for a single block.(6) It analyzes the mechanical behavior of complex multi-block system with the written program. A longcantilever beam with uniform distributed loading and connecting surface is simulated; The influence overdisplacement by distribution, number and stiffness of connecting point is analyzed in detail. Besides, itanalyzes the mechanical behavior of structures assembled by plate and beam with the written program; theresult is compared with that of Finite Element Method. It verifies the convergence, accuracy and simpleness ofhigh order DDA method.