A natural extension of the conventional finite volume method into polygonal unstructured meshes for CFD application.

A new general cell-centered solution procedure based upon the conventional control or finite volume (CV or FV) approach has been developed for numerical heat transfer and fluid flow which encompasses both structured and unstructured meshes for any kind of mixed polygon cell. Unlike conventional FV methods for structured and block structured meshes and both FV and FE methods for unstructured meshes, the irregular control volume (ICV) method does not require the shape of the element or cell to be predefined because it simply exploits the concept of fluxes across cell faces. That is, the ICV method enables meshes employing mixtures of triangular, quadrilateral, and any other higher order polygonal cells to be exploited using a single solution procedure. The ICV approach otherwise preserves all the desirable features of conventional FV procedures for a structured mesh; in the current implementation, collocation of variables at cell centers is used with a Rhie and Chow interpolation (to suppress pressure oscillation in the flow field) in the context of the SIMPLE pressure correction solution procedure. In fact all other FV structured mesh-based methods may be perceived as a subset of the ICV formulation. The new ICV formulation is benchmarked using two standard computational fluid dynamics (CFD) problems i.e., the moving lid cavity and the natural convection driven cavity. Both cases were solved with a variety of structured and unstructured meshes, the latter exploiting mixed polygonal cell meshes. The polygonal mesh experiments show a higher degree of accuracy for equivalent meshes (in nodal density terms) using triangular or quadrilateral cells; these results may be interpreted in a manner similar to the CUPID scheme used in structured meshes for reducing numerical diffusion for flows with changing direction.

An extension of Purcell's vector method with applications to panel element equations

The classical Purcell's vector method, for the construction of solutions to dense systems of linear equations is extended to a flexible orthogonalisation procedure. Some properties are revealed of the orthogonalisation procedure in relation to the classical Gauss-Jordan elimination with or without pivoting. Additional properties that are not shared by the classical Gauss-Jordan elimination are exploited. Further properties related to distributed computing are discussed with applications to panel element equations in subsonic compressible aerodynamics. Using an orthogonalisation procedure within panel methods enables a functional decomposition of the sequential panel methods and leads to a two-level parallelism.

A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate

A novel multi-scale seamless model of brittle-crack propagation is proposed and applied to the simulation of fracture growth in a two-dimensional Ag plate with macroscopic dimensions. The model represents the crack propagation at the macroscopic scale as the drift-diffusion motion of the crack tip alone. The diffusive motion is associated with the crack-tip coordinates in the position space, and reflects the oscillations observed in the crack velocity following its critical value. The model couples the crack dynamics at the macroscales and nanoscales via an intermediate mesoscale continuum. The finite-element method is employed to make the transition from the macroscale to the nanoscale by computing the continuum-based displacements of the atoms at the boundary of an atomic lattice embedded within the plate and surrounding the tip. Molecular dynamics (MD) simulation then drives the crack tip forward, producing the tip critical velocity and its diffusion constant. These are then used in the Ito stochastic calculus to make the reverse transition from the nanoscale back to the macroscale. The MD-level modelling is based on the use of a many-body potential. The model successfully reproduces the crack-velocity oscillations, roughening transitions of the crack surfaces, as well as the macroscopic crack trajectory. The implications for a 3-D modelling are discussed.

Multiscale numerical modelling of crack propagation in two-dimensional metal plate

A novel multiscale model of brittle crack propagation in an Ag plate with macroscopic dimensions has been developed. The model represents crack propagation as stochastic drift-diffusion motion of the crack tip atom through the material, and couples the dynamics across three different length scales. It integrates the nanomechanics of bond rupture at the crack tip with the displacement and stress field equations of continuum based fracture theories. The finite element method is employed to obtain the continuum based displacement and stress fields over the macroscopic plate, and these are then used to drive the crack tip forward at the atomic level using the molecular dynamics simulation method based on many-body interatomic potentials. The linkage from the nanoscopic scale back to the macroscopic scale is established via the Ito stochastic calculus, the stochastic differential equation of which advances the tip to a new position on the macroscopic scale using the crack velocity and diffusion constant obtained on the nanoscale. Well known crack characteristics, such as the roughening transitions of the crack surfaces, crack velocity oscillations, as well as the macroscopic crack trajectories, are obtained.

A multi-scale numerical modelling of crack propagation in a 2D metallic plate

A new multi-scale model of brittle fracture growth in an Ag plate with macroscopic dimensions is proposed in which the crack propagation is identified with the stochastic drift-diffusion motion of the crack-tip atom through the material. The model couples molecular dynamics simulations, based on many-body interatomic potentials, with the continuum-based theories of fracture mechanics. The Ito stochastic differential equation is used to advance the tip position on a macroscopic scale before each nano-scale simulation is performed. Well-known crack characteristics, such as the roughening transitions of the crack surfaces, as well as the macroscopic crack trajectories are obtained.

Finite element modelling of crack detection tests

Four non-destructive tests for determining the length of fatigue cracks within the solder joints of a 2512 surface mount resistor are investigated. The sensitivity of the tests is obtained using finite element analysis with some experimental validation. Three of the tests are mechanically based and one is thermally based. The mechanical tests all operate by applying different loads to the PCB and monitoring the strain response at the top of the resistor. The thermal test operates by applying a heat source underneath the PCB, and monitoring the temperature response at the top of the resistor. From the modelling work done, two of these tests have shown to be sensitive to cracks. Some experimental results are presented but further work is required to fully validate the simulation results.

Assessing the performance of crack detection tests for solder joints

This paper presents both modelling and experimental test data to characterise the performance of four non-destructive tests. The focus is on determining the presence and rough magnitude of thermal fatigue cracks within the solder joints for a surface mount resistor on a strip of FR4 PCB. The tests all operate by applying mechanical loads to the PCB and monitoring the strain response at the top of the resistor. The modelling results show that of the four tests investigated, three are sensitive to the presence of a crack in the joint and its magnitude. Hence these tests show promise in being able to detect cracking caused by accelerated testing. The experimental data supports these results although more validation is required.

Computer simulation of crack propagation in power electronics module solder joints

A numerical modelling method for the analysis of solder joint damage and crack propagation has been described in this paper. The method is based on the disturbed state concept. Under cyclic thermal-mechanical loading conditions, the level of damage that occurs in solder joints is assumed to be a simple monotonic scalar function of the accumulated equivalent plastic strain. The increase of damage leads to crack initiation and propagation. By tracking the evolution of the damage level in solder joints, crack propagation path and rate can be simulated using Finite Element Analysis method. The discussions are focused on issues in the implementation of the method. The technique of speeding up the simulation and the mesh dependency issues are analysed. As an example of the application of this method, crack propagation in solder joints of power electronics modules under cyclic thermal-mechanical loading conditions has been analyzed and the predicted cracked area size after 3000 loading cycles is consistent with experimental results.

Singular stresses at the tip of a crack terminating at frictional interface of fibre reinforced composites

The stress singularities at the tip of a crack that terminates at a frictional interface between two layers in anisotropic composites are investigated. The order of stress singularities is determined by solving the characteristic equations obtained from the boundary conditions and the frictional interface conditions for the cases concerned. The interface is assumed to be governed by Coulomb's law of friction. Numerical results are presented for the cases with a crack terminating at a frictional interface of a fibre reinforced composite, and it is shown that there is a big difference of stress singularities between cases with and without considering friction along the interface.