Wavelet based Adaptive RBF Method for Nearly Singular Poisson-Type Problems on Irregular Domains

We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined over Ω ∈ ℜd, the boundary of an irregularly shaped domain, Γ, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition level dependent thresholding were introduced to improve the efficiency and convergence rate of the solution. We will show how the generalized wavelet based adaptive method can be applied for detecting nearly singularities in Poisson type PDEs over irregular domains. The numerical examples have illustrated that the proposed method is powerful to analyze the Poisson type PDEs with rapid changes in gradients and nearly singularities

Stable PDE Solution Methods for Large Multiquadric Shape Parameters

We present a new method based upon the paper of Volokh and Vilney (2000) that produces highly accurate and stable solutions to very ill-conditioned multiquadric (MQ) radial basis function (RBF) asymmetric collocation methods for partial differential equations (PDEs). We demonstrate that the modified Volokh-Vilney algorithm that we name the improved truncated singular value decomposition (IT-SVD) produces highly accurate and stable numerical solutions for large values of a constant MQ shape parameter, c, that exceeds the critical value of c based upon Gaussian elimination

A Fast Adaptive Wavelet Scheme in RBF Collocation for Nearly Singular Potential PDEs

We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs. Multiresolution wavelet analysis (MRWA) provides a firm mathematical foundation by projecting the solution of PDE onto a nested sequence of approximation spaces. The wavelet coefficients then were used as an estimation of the sensible regions for node adaptation. The proposed adaptation scheme requires negligible calculation time due to the existence of the fast DiscreteWavelet Transform (DWT). Certain aspects of the proposed adaptive scheme are discussed through numerical examples. It has been shown that the proposed adaptive scheme can detect the singularities both in the domain and near the boundaries. Moreover, the proposed adaptive scheme can be utilized for capturing the regions with high gradient both in the solution and its spatial derivatives. Due to the simplicity of the proposed method, it can be efficiently applied to large scale nearly singular engineering problems

A Stabilized RBF Collocation Scheme for Neumann Type Boundary Value Problems

The numerical solution of partial differential equations (PDEs) with Neumann boundary conditions (BCs) resulted from strong form collocation scheme are typically much poorer in accuracy compared to those with pure Dirichlet BCs. In this paper, we show numerically that the reason of the reduced accuracy is that Neumann BC requires the approximation of the spatial derivatives at Neumann boundaries which are significantly less accurate than approximation of main function. Therefore, we utilize boundary treatment schemes that based upon increasing the accuracy of spatial derivatives at boundaries. Increased accuracy of the spatial derivative approximation can be achieved by h-refmement reducing the spacing between discretization points or by increasing the multiquadric shape parameter, c. Increasing the MQ shape parameter is very computationally cost effective, but leads to increased ill-conditioning. We have implemented an improved version of the truncated singular value decomposition (IT-SVD) originated by Volokh and Vilnay (2000) that projects very small singular values into the null space, producing a well conditioned system of equations. To assess the proposed refinement scheme, elliptic PDEs with different boundary conditions are analyzed. Comparisons that made with analytical solution reveal superior accuracy and computational efficiency of the IT-SVD solutions

Quantitative phase-field modeling of crack propagation in multi-phase materials

”Research presented in this dissertation is focused on developing and validating a computational framework for study of crack propagation in polycrystalline composite ceramics capable of designing micro-architectures of phases to improve fracture toughness and damage tolerance of ZrB2-based ultra-high temperature ceramics (UHTCs). A quantitative phase-field model based on the regularized formulation of Griffith’s theory is presented for crack propagation in homogenous and heterogeneous brittle materials. This model utilizes correction parameters in the total free energy functional and mechanical equilibrium equation within the crack diffusive area to ensure that the maximum stress in front of the crack tip is equal to the stress predicted by classical fracture mechanics. Also, unlike other phase-field models, the effect of material strength on crack nucleation and propagation was considered. The accuracy of the model is benchmarked in different ways and the simulation results are validated against experimental results for concrete in the form of fracture of L-shaped plates and wedge splitting tests, and for ZrB2-based laminates and fibrous monolithic composites. To study crack propagation in polycrystalline systems, a phase-field model for grain growth is coupled to the proposed model for crack propagation in multi-phase systems. Intergranular and transgranular crack propagation in ZrB2-bicrystal and polycrystalline systems in mode-I loading are studied. The significant advantages of the proposed model are revealed in multi-phase systems with considerably different material properties for different phases in which the model enables accurate predication of the crack propagation path in composites consisting of materials with significantly different strengths”--Abstract, page iii

Phase-Field Modeling of Crack Propagation in Polycrystalline Materials

A phase-field model based on a modified form of the regularized formulation of Griffith\u27s fracture theory is presented to investigate intergranular and transgranular crack propagations in polycrystalline brittle materials. Grains and grain boundaries are incorporated in the crack initiation and propagation model based on a phase-field model for grain growth, in which the elastic anisotropy varies based on the grain orientation angle, and the grain boundary energy is related to the misorientation angle of the adjacent grains. Correction parameters are utilized in the total free energy functional and mechanical equilibrium equations to consider the effect of material strength on crack nucleation and propagation independent of the regularization parameter. This allows controlling the strength and crack surface energy along the grain boundaries as a function of the misorientation angle in order to mediate intergranular and/or transgranular crack propagation. To demonstrate the capability of the proposed model, intergranular and transgranular crack propagation in ZrB2 bicrystal systems under tensile loading are studied in detail. The effects of grain boundary misorientation angle, grain boundary inclination with respect to initial crack direction, and grain boundary strength (and/or crack surface energy) on the crack propagation path are investigated. Intergranular crack propagation can be promoted by specific combinations of grain boundary strength and crack surface energy, which can contribute to the fracture toughness of polycrystalline materials

Science strategy England's Northwest

Title from coverAvailable from British Library Document Supply Centre- DSC:m03/18771 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo

Predicting Effective Fracture Toughness of ZrB₂-Based Ultra-High Temperature Ceramics by Phase-Field Modeling

The effective fracture toughness (EFT) of ZrB2-C ceramics with different engineered microarchitectures was numerically evaluated by phase-field modeling. To verify the model, fibrous monoliths (elongated hexagonal ZrB2-rich cells in a continuous C-rich matrix) with different volume fractions of a C-rich phase were considered. Architectures containing 10 and 30 vol% of C-rich phase showed EFT values about 42% more than that of pure ZrB2. Increasing the C-rich phase to 50 vol%, dropped toughness significantly, which is in agreement with the experimental results. Replacing hexagonal cells with cylindrical, triangular, or square cells of the same cross-sectional area changed the toughening mechanism and EFT. The orientation of the interface between the soft and hard phases with respect to the crack orientation also affected the energy required for crack propagation, and in some cases resulted in a higher EFT (even up to 70% of pure ZrB2 fracture toughness) either by suppressing uniform crack propagation or making crack cranking. Results not only show that the model can predict fracture toughness but also provide insight to improve toughness by engineering different microarchitectures

A Modified Phase-Field Model for Quantitative Simulation of Crack Propagation in Single-Phase and Multi-Phase Materials

A quantitative phase-field model based on the regularized formulation of Griffith\u27s theory is presented for crack propagation in homogenous and heterogeneous brittle materials. This model utilizes correction parameters in the total free energy functional and mechanical equilibrium equation in the diffusive crack area to ensure that the maximum stress in front of the crack tip is equal to the stress predicted by classical fracture mechanics. Also, unlike other phase-field models, the effect of material strength on crack nucleation and propagation was considered independent of the regularization parameter. The accuracy of the model was benchmarked in two ways. First, the stress and strain fields around the crack tip in single-phase ZrB2 were compared with the analytical solutions in classical linear elastic fracture mechanics. Second, the crack path and force—displacement responses were examined against experimental results for concrete in the form of fracture of L-shaped plate and wedge splitting tests. To demonstrate the capability of the model in multi-phase materials, crack propagation was simulated for laminates composed of alternating layers of ZrB2 and carbon. The results showed that the proposed modifications in the phase-field model were necessary to predict crack deflection along carbon layers similar to the experimental observations

A Numerical Comparison of Different Approximation Techniques in Meshfree Methods

There is a great interest in applying numerical methods for solution of partial differential equations of various types of engineering problems. Numerical solution of partial differential equation consist of two main parts: approximating the solution of equation (or unknown function) then determining the unknown function by embedding approximated solution in the governing equation in the weak or strong form and applying boundary conditions to determine the unknown function. Accuracy and efficiency of numerical method is strictly dependent to accuracy and efficiency of these two parts. During last decade, various methods have been evolved rapidly and efficient methods such as EFG, SPH, PIM, MFS and RBF were successfully used for different engineering problems. In this paper we compare accuracy and efficiency of different techniques used in meshfree methods which proposed and used by researchers for approximating the solution of equation. Results show that there is a great difference between accuracy and time efficiency of these meshfree methods. Some numerical example in elasticity problems were studied in this paper. Stress and displacement approximation of a cracked media also studied in this paper to reveal the efficiency of mesh free methods in approximating singular functions and its derivatives