Treatment Of Domain Integration In Boundary Face Method And Its Application In Elastic-dynamics

With the rapid development of computer hardware,the role of CAE analysis in engineering applications is becoming more and more important.The finite element method is developing rapidly and widely used,but the finite element method itself has some defects.With the development of computer,the boundary integral method equation has been perfected,with the advantages of dimensionality reduction and high calculation accuracy.The boundary face method inherits all the advantages of the boundary integral equation method,and performs discrete analysis calculation directly on the 3D solid model.It does not simplify the model,thus avoiding geometric errors.It is a method of CAD/CAE integrated and Isogeometric analysis.In the implementation of the boundary face method,the boundary integral and the domain integral have a great influence on the calculation accuracy.Therefore,this paper will focus on the singular integral and near singular integral,domain integral and Gaussian integral criterion in the boundary face method,and propose some solutions to solve them.This will broaden the application of the boundary face method in engineering.The main research work and results of th is paper are as follows:(1)A method for solving weak singular integrals is proposed.Based on the weak singularity in the basic solution,a four-node serendipity triangular integral sub-element is proposed.On this basis,a simpler new(?,?)coordinate transformation is introduced to eliminate the weak singularity of the fundamental solution.One side of this four-node serendipity triangular sub-element is a quadratic curve,the other two sides are straight lines,and the distance from the source point t o the three nodes on the curve is equal.Then,the reason why the integral sub-elements with large angles at the source point position have near singularity in the direction of the ring is analyzed.In order to overcome the integral problem of the hoop dir ection,through the study of the position of the middle node of the triangular integral patch,through a large number of numerical experiments,the optimal position of the middle node is found,so that the final integral accuracy and efficiency are the highest,and then a class resolution method for solving weak singular integrals is obtained,and the corresponding theoretical derivation process is given.(2)A solution to two-dimensional and three-dimensional domain integration and corresponding equal-precision Gaussian integration principal are proposed.Aiming at the near-singular domain integral,a quadtree element adaptive subdivision method is proposed.The method determines whether the subdivision is subdivided by the distance from the source point to the element center and the element side length.The closer the source point is to the element,the finer.The more sub-elements are dropped,the smaller the side length of the sub-elements near the source point.For the singular domain integration,a method of adaptive spherical subdivision is proposed.The method can be applied to any element shape and can separate sub-elements whose shape is favorable for integration.Then the method of solving the singular integral by two-dimensional class analysis is extended to three-dimensional.Although the analytical calculation cannot be guaranteed,the Gaussian integration point can be effectively reduced under the same precision.The precision Gaussian integral criterions such as singular and near-singular domain integrals are proposed.It is guaranteed that when these sub-elements are equal accurately integrated,the Gaussian points near the source point are densely distributed,and the Gaussian points away from the source point are sparse.While improving the accuracy,the computational efficiency can also be effectively reduced.(3)An expanding element interpolation method and a near-singular integral processing scheme are proposed and applied to the solution of thin-walled structures.The expanding element is achieved by collocating virtual nodes along the perimeter of the traditional discontinuous element,the internal point is called the source point,and the boundary integral equation is only configured by the source point.The ex panding element retains the advantages of both continuous and non-contiguous elements,while overcoming their shortcomings,and improving the interpolation accuracy by at least two orders without changing the degree of freedom of the equation.Finally,the method of processing near-singular domain integral is applied to solve the near-singular integral of thin-walled structure.The integral element is subdivided into two identical sub-elements according to the position of the source point.The source point is closer to the element,the more sub-elements are subdivided.Through this method,the near-singular integral can be accurately calculated.(4)Aiming at the instability of time domain method,a singular integral subdivision technique related to time step is proposed to improve the calculation accuracy of elastic dynamic time domain method.Based on the ex panding element,a new type double-layer interpolation is adopted to interpolate the physical quantities in the integral equation,can make the wave front to get a better simulation,improve the interpolation accuracy of the dynamic response at the wave front,and thus improve the final calculation accuracy and stability.In addition,based on the boundary face method,the discrete solution of time and space for the two-dimensional elastodynamics boundary integral equation is realized.(5)The pseudo-intial condition method of three-dimensional transient elastodynamics is implemented.An equivalent pseudo force method is used to deal with the non-zero initial condition in the integra l equation,and the corresponding initial condition is equivalent to the pseudo force.The domain integrals of the initial condition item have to be solved.The pseudo-initial condition method regards the calculation result in the domain of the previous st ep as the initial condition of the current step,which is easy to cause the accumulation of errors.For this problem,the scheme of processing domain integral is improved,and the calculation error caused by the introduction of domain integral is reduced.And calculation accuracy can be indirectly improved.The numerical implementation of the problem of elastodynamics under arbitrary initial conditions and arbitrary body forces is realized by the boundary face method.