Research On Nearly Singular Integral Technique And Element Interpolation Method In Boundary Face Method

Boundary element method(BEM)has been widely applied to various research fields of engineering and scientific problems.This is due to its unique advantages:higher accuracy,the reduction of the dimensionality of the problem by one,and naturally solving both singularity problems and infinite domain problems.In BEM analysis,the conventional Lagrange elements are employed for the characterization of both geometric and physical variables.Obviously,this leads to introduce geometric errors and reduce computational accuracy.Boundary face method(BFM)is also theoretically based on the boundary integral equation,but it is implemented directly on the CAD model.Thus,BFM not only inherits all advantages of BEM,but also avoids the geometric errors,thereby naturally realizing seamless integration of CAD and CAE.In BFM analysis,a large number of elements are required to obtain an acceptable engineering accuracy,especially for structures with small features and thin walls.Thus,this leads to extremely expensive computational costs.Two key factors could account for this phenomenon.First,both singular integrals and nearly singular integrals are widely existed in these structures,greatly affecting the performance of the BFM.Second,the order of the interpolation function of conventional Lagrange element is lower.This thesis is devoted to the research of singular integral techniques,nearly singular integral techniques and element interpolation methods in BFM,and applied to solve potential problems,elasticity problems and acoustic problems of structures with small features and thin walls.The main contents are as follows:(1)Research on nearly singular integral technique based on nonlinear transformation method.According to the nearest point from a source node to an integration element,a new distance function,the nearest distance function,is proposed.This distance function is able to unify the traditional normal and tangential distance functions.Based on the new function,two nonlinear transformation formulas are proposed.By combining the nonlinear transformation method and element subdivision method,a solution of the nearly weakly and strongly singular integrals is presented.By combining the nonlinear transformation method,element subdivision method and virtual boundary element method,using the properties of the hyper-singular integrals,a solution of the nearly hyper-singular integrals is presented.These schemes has been successfully applied to evaluate nearly weakly,strongly and hyper-singular integrals,and solve the problems of thin-walled structures.In theses schemes,the effective proximity of the nearly integrals can be reduced to 10^-14.(2)Research on nearly singular integral technique based on semi-analytical method.A semi-analytical method based on the nearest point is proposed.By this method,the solutions of nearly singular integrals based on nonlinear transformation method are improved.Numerical results demonstrate that,compared to other nearly singular integral methods,this algorithm is able to provide higher accuarcy and efficiency.Moreover,this algorithm is able to overcome the problem of the nonlinear transformation method.When the source node is extremely close to the integral element,the difference of the numerical integral values among the integration subelements is large.Obivuously,the data rounding errors are introduced and thereby the computational errores of the nearly singular integrals are increased.(3)Research on dual interpolation method(DIM)based on meshless interpolation method.The DIM for potential problems is developed.In this method,the improved interpolation moving least-square(IIMLS)method is used to establish the second layer interpolation of DIM.The DIM has the following advantages:unifying the traditional continuous and discontinuous elements,improving the order of interpolation functions of discontinuous elements and naturally approximating continuous and discontinuous fields.Based on this method,the dual interpolation boundary face method(Di BFM)for potential problems is developed,and successfully applied to solve two-dimensional steady-state heat conduction problems whose domains contain small features and thin walls.Numerical results demonstrate that,compared to BFM and finite element method(FEM),Di BFM possesses higher accuracy,efficiency,and convergence rates.(4)Research on DIM based on meshless interpolation method,element interpolation and physical relationship.A new DIM is developed,in which the second layer interpolation is established using the moving least-squares(MLS)method,element interpolation method and physical relationship.The main difference between the two DIMs is that:even if a few elements are placed on the short edges and small features,the new DIM can still improve the order of interpolation functions of the elements,thus ensuring the performance of the Di BFM.Based on the new DIM,the Di BFM for elasticity problems is developed,and successfully applied to stress analysis of two-dimensional structures with small features and thin walls.Numerical results demonstrate that,compared to BFM,Di BFM of(3)and FEM,this method possesses higher accuracy,efficiency,and convergence rates.Moreover,it has the ability to accurately compute the stress concentration around small corners with relatively coarse discretization.(5)Research on Di BFM for acoustic problems based on Burton-Miller formulation.Extending the DIM to acoustic problems,the Di BFM for acoustic problems is developed.The Burton-Miller formulation is empolyed to solve the non-uniqueness of the exterior acoustic problems.To address the oscillation of various kernel functions in acoustic problems,an adaptive singular integral algorithm is developed.This algorithm can also be applied to evaluate the weakly,strongly and hyper-singular integrals in potential and elasticity problems.Numerical results demonstrate that Di BFM is suitable for solving mid-frequency acoustic problems.(6)Research on BFM based on expanding element interpolation method.An expanding element interpolation method for three-dimensional problems is developed,and successfully applied to slove elasticity problems whose domains contain small features.In this method,both element interpolations and physical relationships are used to establish the relationships between virtual nodes and source nodes.Unlike the DIM based on MLS,the main advantage of this method is that there is no need to judge the number of interpolation nodes in each interpolation direction.Therefore,it is suitable for solving the problems of structures with narrow and irregular cut surfaces.Numerical examples demonstrate the effectiveness of this method.