In this paper a collocation approach is presented for spatial discretization of the partial integrodifferential equation arising in a peridynamic formulation in stochastic fracture mechanics. In the formulation nodes are distributed inside the domain forming a grid, and the inverse multiquadric radial basis functions are used as interpolation functions inside the domain. Due to this discretization the peridynamic stiffness is generated in a manner similar to the finite element method. Further, any discontinuity in the domain is included in this discretized form and affects only the peridynamic stiffness of the adjacent nodes. Using this approach as a tool, the probability density function of the energy release rate can be determined at a given crack tip point for all possible crack paths. Thus, the crack propagation direction can be probabilistically identified. This is accomplished by numerical evaluation of the requisite Neumann expansion using pertinent Monte Carlo simulations. Specific examples of applications are included.
Significant research on bottom-hole-assembly (BHA) vibrations in conjunction with continuous improvement in polycrystalline diamond cutter (PDC) bits using fluid mechanics has increased drilling efficiency and reduced related problems. Despite the analytical advances, most BHAs and bit – motor combinations are still heuristically chosen and most optimization efforts are by trial and error. The reason for this is the bit-rock interaction and the dynamics that the bit exerts on the BHA are stochastic processes, and most analyses use simplified assumptions for this complicated interaction. In this paper, numerical solutions for the differential equations describing the motion and dynamics of the BHA are provided including impact effects with the borehole while drilling near vertical wellbores. Secondly, the dynamics of this motion are coupled with a neural network model that is calibrated on previous bit runs. This approach renders significant improvements in predicting the performance of different bit-motor combinations and provides a reasonable estimation of ROP without having to actually run those combinations and re-calibrate the model. The main reasons why a strictly deterministic, numerical approach cannot be successful is that even though the BHA properties are well known and vibration characteristics can be estimated, the impact forces of those components with an imperfect bore hole with variable formation properties cannot be precisely calculated. In addition, the forces exciting the BHA – drillstring system through the bit are complex, multi-modal and semi-chaotic.
The neural network is hence used to identify and calibrate a model of the bit-rock interaction, essentially acting to partially de-couple BHA effects from how each bit achieves ROP. Several parameters affecting the drilling process are used as input and help uniquely identify the bit-rock interaction for a range of drilling parameters. This approach allows for a reasonable extrapolation of the bit’s behaviour on a different BHA and with different drilling parameters.
In this paper a non linear viscoelastic model governed by fractional derivatives is presented for modeling the in-service behavior of polyester mooring lines. In the formulation an iterative approach utilizing the Gauss-Newton minimization algorithm in conjunction with the catenary equations used to determine the static modulus of elasticity and the effective length of polyester mooring lines corresponding to calm sea conditions. Upon establishing the accuracy of the static modulus via comparison with field data, the catenary equations and the offshore platform’s position versus time are used to identify the polyester strain under developed-sea conditions. In this manner, time histories of stress and strain for polyester ropes in service conditions are obtained. Then, a non linear viscoelastic model involving fractional derivative terms is used to capture the in service polyester line behavior. For this, the tension of the proposed model corresponding to the actual polyester strain is compared at each time step to the tension obtained from the field data. Finally, the parameters of the proposed model are derived by minimizing the error in the least-squares sense over a large number of data points using the Levenberg-Marquardt algorithm. The numerically derived force-strain relationship is found to be in reasonable agreement with supplementary field and laboratory experimental data, the field data pertain to an offshore structure moored in position using polyester mooring lines operated in the Gulf of Mexico during Hurricane Katrina (August of 2005).
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