Research On Boundary Element Method Of B-spline Wavelet On The Interval

In recent years,finite element method(FEM)of B-spline wavelet on the interval(BSWI)develops rapidly,has became one of the powerful tool to solve various engineering and scientific problems.BSWI FEM possesses two prominent advantages.One is to achieve multi-scale approximation by directly using scaling functions at different scale.While the other one is that the computational efficiency and precision can be further improved because of the excellent approximation property of wavelet bases.However,the only construction method of tensor product wavelet is difficult to be applied to irregular computational domains in high dimensional problems.Comparing with FEM,boundary element method(BEM)embodies advantages of dimensionality reduction and simple mesh generation.Consequently,it is necessary to combine BEM and BSWI to investigate BSWI BEM.Aiming at four common engineering problems to study BSWI BEM,they are potential problem,elasticity problem,acoustic problem and optical problem.Based on the corresponding one dimensional and two dimensional boundary integral equations of two dimensional and three dimensional potential problems,the BSWI scaling functions are applied to derive the calculated formats of BSWI BEM,construct BSWI elements,discretize geometric shape and form BSWI BEM algebraic equations to solve potential problems,typical examples are provided to verify the efficiency of the present method through comparing with conventional BEM and analytical solutions.According to the corresponding one dimensional and two dimensional boundary integral equations of two dimensional and three dimensional elasticity problems,using the similar approaches with potential problems to give computational formats of BSWI BEM,construct BSWI elements,divide geometric boundary as well as generate BSWI BEM algebraic equations to deal with elasticity problems.In addition,the effectiveness of BSWI BEM is testified by the presented examples.For two dimensional acoustic and optical problems,the BSWI BEM algebraic equations about matrix and scatterer of two dimensional photonic and phononic crystal are established in a unit cell based on the general Green's function and BSWI elements.In order to illustrate the general applicability of BSWI BEM in solving two dimensional acoustic and optical problems,the BSWI BEM models used for calculating band structure of photonic and phononic crystal arranged in a square and triangular type are developed through combing with Bloch theory and interface connection conditions between matrix and scatterer.Some provided numerical examples show that the BSWI BEM have competitive edge and general applicability.