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The coupled equations for a propagating hydraulic fracture (HF) exhibit a multi-scale structure for which resolution on all length scales becomes computationally prohibitive. Asymptotic analysis is able to identify the dominant physical process active at the computational length scale. This paper describes a novel algorithm that uses weak-form tip asymptotics on a rectangular Eulerian mesh to solve the problem of a propagating HF. The location of the fracture front is determined within each tip element by matching the volume associated with the known asymptotic solution to the flux of fluid into the given element. Even if the fracture front is curved, the algorithm is able to capture the solution on a relatively coarse rectangular mesh by implementing a weak form of the tip asymptotic solution, based on averaging the volume over a tip element to determine the fracture widths at tip element centers. The fracture is divided into a ‘channel’ region made up of elements that are filled with fluid and a ‘tip’ region comprising partially filled elements. An iterative procedure is used to determine the fracture widths and fluid pressures in the channel region and the fracture font locations in the tip regions. The algorithm is tested by analyzing an HF propagating in a viscosity-dominated regime within an impermeable homogeneous elastic material for which a similarity solution is available. The numerical results show close agreement with the exact solution even though a relatively coarse mesh is used. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper describes a finite element solution algorithm for the numerical solution of large size three-dimensional flow problems on a personal computer. To demonstrate the algorithm, the Stokes equations in velocity–vorticity form are solved for a lid-driven cubical cavity problem. The Galerkin's weighted residual form of the governing equations is evaluated for all the elements of the computational domain and kept as element-matrices and element-vectors. This results in a set of simultaneous equations corresponding to the global nodes of each element. Those elements that contain the boundary nodes are modified to incorporate the Dirichlet boundary conditions. A conjugate gradient iterative scheme is employed to solve the simultaneous equations in element form to get the solution at the global nodes. The matrix–vector products used in the conjugate gradient iterative solver are performed in element level, assembling only the element-level vectors to form the global vectors. Since the element-level computation has eluded the formation of global matrices, the numerical solution of three-dimensional Stokes equations using a mesh of size as high as 513 could be achieved on a personal computer. The algorithm is validated by comparing the results for a three-dimensional transient diffusion problem and Stokes flow in a lid-driven cubical cavity. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper introduces an elastic predictor/return mapping integration algorithm for a simplified version of the Lemaitre ductile damage model, whose return mapping stage requires the solution of only one scalar non-linear equation. The simplified damage model differs from its original counterpart only in that it excludes kinematic hardening. It can be used to predict ductile damage growth whenever load reversal is absent or negligible—a condition met in a vast number of practical engineering applications. The one-equation integration scheme proves highly efficient in the finite element solution of typical boundary value problems, requiring computation times comparable to those observed in classical von Mises implementations. This is in sharp contrast to the previously proposed implementations of the original model whose return mapping may require, in the most general case, the solution of a system of 14 coupled algebraic equations. For completeness, a closed formula for the corresponding consistent tangent operator is presented. The performance of the algorithm is illustrated by means of a numerical example. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper describes the incorporation of diffusive transport into the original semi-Lagrangian DISCUS algorithm for pure advection. An explicit treatment of diffusion is adopted following the approach used in the QUICKEST algorithm for advection–diffusion. The semi-Lagrangian treatment of the advection term relaxes the small time step restriction normally associated with Eulerian treatments of advection, but the Eulerian treatment of the diffusion term imposes conventional limitations on the scheme. Numerical experiments of advection–diffusion, however, indicate that DISCUS has advantages over the QUICKEST scheme for advection–diffusion in three key areas: stability, accuracy and computational efficiency. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Numerical simulation of complex flows of viscoelastic fluids requires us to use efficient algorithms for solving the non-linear governing equations. The stream-tube analysis permits calculation of such flows by considering elementary subdomains in a mapped computational domain where the transformed streamlines are parallel and straight. The consideration of subregions leads us to use optimization procedures. The paper concerns the application of such algorithms to compute main flows in axisymmetric converging geometries, for fluids obeying memory-integral constitutive equations. [ABSTRACT FROM AUTHOR]