Y.Y. Lu

A new implementation of the element free Galerkin method

Abstract Element free Galerkin methods (EFG) are methods for solving partial differential equations with moving least squares interpolants. EFG methods require only nodal data; no element connectivity is needed. In a previous implementation of the EFG method, Lagrange multipliers were used to enforce the essential boundary condition. However, the use of Lagrange multipliers increases the cost of solving the linear algebraic equations. A new implementation is developed based on a modified variational principle in which the Lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded. In addition, weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated. Numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes; the latter makes it very advantageous for modelling progressive cracking.

Crack propagation by element-free Galerkin methods

Abstract Element free Galerkin methods (EFG) are gridless methods for solving partial differential equations which employ moving least square interpolants for the trial and test functions. EFG methods require only nodes and a description of the external and internal boundaries and interfaces of the model; no element connectivity is needed. The implementation of EFG to arbitrary crack growth in static problems such as fatigue is described. Numerical examples show that accurate stress intensity factors can be obtained by this method without any enrichment of the displacement field by a near crack tip singularity and that crack growth can be easily modeled since it requires almost no remeshing.

Fracture and crack growth by element free Galerkin methods

Element free Galerkin (EFG) methods are methods for solving partial differential equations that require only nodal data and a description of the geometry; no element connectivity data are needed. This makes the method very attractive for the modeling of the propagation of cracks, as the number of data changes required is small and easily developed. The method is based on the use of moving least-squares interpolants with a Galerkin method, and it provides highly accurate solutions for elliptic problems. The implementation of the EFG method for problems of fracture and static crack growth is described. Numerical examples show that accurate stress intensity factors can be obtained Without any enrichment of the displacement field by a near-crack-tip singularity and that crack growth can be easily modeled since it requires hardly any remeshing.

Element-free galerkin methods for static and dynamic fracture

Abstract Element-free Galerkin (EFG) methods are presented and applied to static and dynamic fracture problems. EFG methods, which are based on moving least-square (MLS) interpolants, require only nodal data; no element connectivity is needed. The description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks. This makes the method particularly attractive for growing crack problems, since only minimal remeshing is needed to follow crack growth. In moving least-square interpolants, the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point. Numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method.

Element-free Galerkin method for wave propagation and dynamic fracture

Abstract Element-free Galerkin method (EFG) is extended to dynamic problems. EFG method, which is based on moving least square interpolants (MLS), requires only nodal data; no element connectivity is needed. This makes the method particularly attractive for moving dynamic crack problems, since remeshing can be avoided. In contrast to the earlier formulation for static problems by authors, the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions. With this formulation, the stiffness matrix is symmetric and positive semi-definite, and hence the consistency, conergence and stability analyses of time integration remain the same as those in finite element method. Numerical examples are presented to illustrate the performance of this method. The relationship between the element-free Galerkin method and the smooth particle hydrodynamics (SPH) method is also discussed in this paper. Results are presented for some one-dimensional problems and two-dimensional problems with static and moving cracks.

Explicit multi-time step integration for first and second order finite element semidiscretizations

Abstract Explicit multi-time step (subcycling) integration algorithms based on nodal partitions for both first and second order systems are presented. The algorithm is identical to an earlier algorithm of the senior author, but some simplifications have been made in the actual algorithm which make it easier to implement in general purpose programs. Consistency, convergence and stability analyses of this algorithm for first order systems are given. Interpolations on the interfaces between different time step domains are studied. A general conjecture on the stability criterion for second order systems with subcycling and nodal partitions is also proposed, and some unsolved theoretical difficulties are posed.

A variationally coupled finite element-boundary element method

Abstract A variationally based coupling is developed for the finite element and boundary element methods. This is achieved by combining the variational forms for the boundary element and finite element subdomains to obtain a global variational form and then choosing a suitable set of test and trial functions. This method is illustrated in the context of the Laplace equation, and a numerical example is given.

A variationally coupled FE-BE method for elasticity and fracture mechanics

A new method for coupling finite element and boundary element subdomains in elasticity and fracture mechanics problems is described. The essential feature of this new method is that a single variational statement is obtained for the entire domain, and in this process the terms associated with tractions on the interfaces between the subdomains are eliminated. This provides the additional advantage that the ambiguities associated with the matching of discontinuous tractions are circumvented. The method leads to a direct procedure for obtaining the discrete equations for the coupled problem without any intermediate steps. In order to evaluate this method and compare it with previous methods, a patch test for coupled procedures has been devised. Evaluation of this variationally coupled method and other methods, such as stiffness coupling and constraint traction matching coupling, shows that this method is substantially superior. Solutions for a series of fracture mechanics problems are also reported to illustrate the effectiveness of this method.

Convergence and stability analyses of multi-time step algorithm for parabolic systems

Abstract An analysis of the consistency and convergence of a subcycling algorithm together with the stability conditions for parabolic systems of both element and nodal partitions are provided. It is shown that the algorithms only attain a first-order rate-of-convergence for both element and nodal partitions in the sense of the total cycle update. The stability conditions for general integration parameters in an element partition are also given for the first time.

A curvilinear spectral overlay method for high gradient problems

Abstract The spectral overlay method is extended to domains with curvilinear boundaries. In the spectral overlay method, high resolution of localized steep gradients is achieved by overlaying a spectral interpolant on a standard finite element mesh. A special integration scheme is developed which enables the method to satisfy the patch test for a curvilinear overlay. Results are compared to closed from solutions with high gradients and the method is shown to be very powerful in capturing the shape of the gradient field and its peak value.