Accurate Numerical Quadrature Algorithm Based On Peridynamics

Because the spatial derivatives of the partial differential equations do not exist at the crack tip or along the crack surface,the classical continuum mechanics model cannot effectively simulate the natural initiation and dynamic propagation of cracks when solving discontinuous problems.The Peridynamic(PD)model based on the nonlocal theory uses the integral equation of motion to describe the mechanical behavior of material,and its constitutive force function includes the description of damage and fracture,so it is not necessary to carry out the crack path pre-analysis when analyzing the material failure problem.These characteristics make it possible to simulate discontinuous problems with complex failure mechanism and coexistence of multiple cracks.When PD model is solved,the motion equation in the form of integral is usually converted to the form of finite sum.Therefore,it is an important content of the numerical realization of PD model to carry out accurate integration in the horizon of material points.Most actual structures are irregular(especially when they contain defects such as pores and fractures),and their models need to be discretized nonuniformly.At present,the common PD material point integration algorithm cannot accurately calculate the size of the integral region,especially for the non-uniform discrete region,which will reduce the calculation accuracy of the PD model.In this paper,a new integral correction algorithm is proposed to improve the calculation accuracy of PD model,and the effectiveness of the algorithm is verified based on the classical bond based PD model.The main work of this paper includes three parts:(1)By dividing the intersection region between the horizon of PD material point and the adjacent grid into simple subfields,a volume correction algorithm is proposed for the accurate calculation of the intersection volume of the horizon boundary.Then the analytical expressions for calculating intersecting area and intersecting volume in two and three dimensions are derived respectively,and the accurate horizon volume is obtained.(2)The integral domain correction algorithm of PD material point was proposed,and the micro modulus function of PD was reconstructed.The rationality of the scheme was verified by single point integral and Gaussian integral respectively.(3)The geometric centroid of the intersecting region near the horizon boundary was used as the integral point to further improve the overall calculation accuracy,and the validity of the centroid correction algorithm was verified by calculating the strain energy density.In this paper,the m convergence of different discrete mesh sizes is analyzed by simulating the uniaxial tension and transverse shearing of prefabricated central circular plate.By simulating several typical fracture problems and comparing with the experimental results in the literature,the effectiveness of the integral correction algorithm proposed in this paper to improve the calculation accuracy is verified.The simulation results show that:(1)The volume correction algorithm proposed in this paper can accurately calculate the volume of the entire horizon,which is not only applicable to uniform discretization model,but also to non-uniform discretization model.(2)The integral domain correction algorithm proposed in this paper can effectively eliminate the error between the numerical integration realization and the theoretical value.(3)The centroid correction algorithm can effectively improve the accuracy of single-point integration,but the integral correction algorithm combined with Gaussian integration has higher computational accuracy.