Abstract: An improved boundary element method for non-linear viscoelastic flow analysis is reported. In this method, the domain integral representing the non-linear effects is calculated using a more efficient approximating technique. This is achieved by first transforming the domain integral into a form that can be approximated by particular solutions to the original problem. These particular solutions are in fact expressed as a linear combination of radial basis functions, whose coefficients are found by data fitting technique. As a result, the numerical computation of the volume integral is eliminated and a significant reduction of CPU time is achieved. The routines are tested with simple flows and then applied to solve complex three-dimensional direct and inverse extrusion problems of polymeric fluids, such as thermoplastic melts. Inverse extrusion process, where an extrusion die profile needs to be computed for a given profile extrudate, is a very important practical engineering application which is successfully analysed by the present method. Relative to a previous BEM implementation where the volume integral is computed directly using numerical quadratures, a CPU time reduction ranging from 40 to 70% is achieved.
Abstract: This paper reports a neural network (NN) implementation for the numerical approximation of functions of several variables and their first and second order partial derivatives. This approach can result in improved numerical methods for solving partial differential equations by eliminating the need to discretise the volume of the analysis domain. Instead only an unstructured distribution of collocation points throughout the volume is needed. An NN approximation of relevant variables for the whole domain based on these data points is then achieved. Excellent test results are obtained. It is shown how the method of approximation can then be used as part of a boundary element method (BEM) for the analysis of viscoelastic flows. Planar Couette and Poiseuille flows are used as illustrative examples.
Abstract: A new BE-only method is achieved for the numerical solution of viscoelastic flows. A decoupled algorithm is chosen where the fluid is considered as being composed of an artificial Newtonian component and the remaining component which is accordingly defined from the original constitutive equation. As a result the problem is viewed as that of solving for the flow of a Newtonian liquid with the non-linear viscoelastic effects acting as a pseudo-body force. Thus the general solution can be obtained by adding a particular solution (PS) to the homogeneous one. The former is obtained by a BEM for the base Newtonian fluid and the latter is obtained analytically by approximating the pseudo-body force in terms of suitable radial basis functions (RBFs). Embedded in the approximation of the pseudo-body force is the calculation of the polymer stress. This is achieved by solving the constitutive equation using RBF networks (RBFNs). Both the calculations of the PS and the polymer stress are therefore meshless and the resultant BEM-RBF method is a BE-only method. The complete elimination of any structured domain discretisation is demonstrated with a number of flow problems involving the upper convected Maxwell (UCM) and the Oldroyd-B fluids.
Abstract: The indirect boundary element method (IBEM) is formulated for a two-dimensional Stokes flow with moving boundary in the presence of gravity. For simulation of the non-Newtonian fluid flow an iterative scheme for solving a nonlinear system of algebraic equation is used to calculate nonlinear viscous terms. The selected scheme requires values of pseudo forces in each boundary element and each internal collocation point obtained from boundary conditions (for velocities and tractions) and velocity field found at the previous iteration. Boundary element integrals are calculated analytically with a singularity extraction and internal cells integrals are evaluated by using the standard Gaussian quadratures. Internal cells sources are estimated numerically within a finite difference approach to velocity derivatives calculation. The Poiseuille flow of power law fluid in a channel is calculated for a benchmark of the IBEM algorithm. The accuracy and convergence tests are presented for a wide range of power law index (0.2-1.2). In the presence of a free surface, no analytical solution is available. The problem of channel filling with power law fluid is analyzed. The shape and the movement of the free surface are investigated through a special numerical front tracking algorithm based on mesh refinement for the free surface and for the solution domain. Elements of high performance computing are presented. The obtained results coincide with those of other authors. It confirms that the presented algorithm and corresponding software can be applied to study a number of low Reynolds flows, for example in a mould filling technology.
Abstract: The spreading of viscous and viscoelastic fluids on flat and curved surfaces is an important problem in many industrial and biomedical processes. In this work the spreading of a linear viscoelastic fluid with changing rheological properties over flat surfaces is investigated via a macroscopic model. The computational model is based on a macroscopic mathematical description of the gravitational, capillary, viscous, and elastic forces. The dynamics of droplet spreading are determined in sessile and pendant configurations for different droplet extrusion or formation times for a hyaluronic acid solution undergoing gelation. The computational model is employed to describe the spreading of hydrogel droplets for different extrusion times, droplet volumes, and surface/droplet configurations. The effect of extrusion time is shown to be significant in the rate and extent of spreading.
Cites methods from "An improved boundary element method..."
..., boundary element methods for the interface and FEM for the internal nonlinear viscoelastic or inertial terms ....
TL;DR: This well written book is enlarged by the following topics: B-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for theLR and QR algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations and preconditioning techniques.
Abstract: This well written book is enlarged by the following topics:
$B$-splines and their computation, elimination methods for
large sparse systems of linear equations, Lanczos algorithm for
eigenvalue problems, implicit shift techniques for the $LR$ and
$QR$ algorithm, implicit differential equations, differential
algebraic systems, new methods for stiff differential
equations, preconditioning techniques and convergence rate of
the conjugate gradient algorithm and multigrid methods for
boundary value problems. Cf. also the reviews of the German
Abstract: 1 Introduction.- 2 The Boundary Element Method for Equations ?2u = 0 and ?2u = b.- 2.1 Introduction.- 2.2 The Case of the Laplace Equation.- 2.2.1 Fundamental Relationships.- 2.2.2 Boundary Integral Equations.- 2.2.3 The Boundary Element Method for Laplace's Equation.- 2.2.4 Evaluation of Integrals.- 2.2.5 Linear Elements.- 2.2.6 Treatment of Corners.- 2.2.7 Quadratic and Higher-Order Elements.- 2.3 Formulation for the Poisson Equation.- 2.3.1 Basic Relationships.- 2.3.2 Cell Integration Approach.- 2.3.3 The Monte Carlo Method.- 2.3.4 The Use of Particular Solutions.- 2.3.5 The Galerkin Vector Approach.- 2.3.6 The Multiple Reciprocity Method.- 2.4 Computer Program 1.- 2.4.1 MAINP1.- 2.4.2 Subroutine INPUT1.- 2.4.3 Subroutine ASSEM2.- 2.4.4 Subroutine NECMOD.- 2.4.5 Subroutine SOLVER.- 2.4.6 Subroutine INTERM.- 2.4.7 Subroutine OUTPUT.- 2.4.8 Results of a Test Problem.- 2.5 References.- 3 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y).- 3.1 Equation Development.- 3.1.1 Preliminary Considerations.- 3.1.2 Mathematical Development of the DRM for the Poisson Equation.- 3.2 Different f Expansions.- 3.2.1 Case f = r.- 3.2.2 Case f = 1+ r.- 3.2.3 Case f = 1 at One Node and f = r at Remaining Nodes.- 3.3 Computer Implementation.- 3.3.1 Schematized Matrix Equations.- 3.3.2 Sign of the Components of r and its Derivatives.- 3.4 Computer Program 2.- 3.4.1 MAINP2.- 3.4.2 Subroutine INPUT2.- 3.4.3 Subroutine ALFAF2.- 3.4.4 Subroutine RHSVEC.- 3.4.5 Comparison of Results for a Torsion Problem using Different Approximating Functions.- 3.4.6 Data and Output for Program 2.- 3.5 Results for Different Functions b = b(x,y).- 3.5.1 The Case ?2u = ?x.- 3.5.2 The Case ?2u = ?x2.- 3.5.3 The Case ?2u = a2 ? x2.- 3.5.4 Results using Quadratic Elements.- 3.6 Problems with Different Domain Integrals on Different Regions.- 3.6.1 The Subregion Technique.- 3.6.2 Integration over Internal Region.- 3.7 References.- 4 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u).- 4.1 Introduction.- 4.2 The Convective Case.- 4.2.1 Results for the Case ?2u = ??u/?x.- 4.2.2 Results for the Case ?2u = ?(?u/?x+ ?u/?y).- 4.2.3 Internal Derivatives of the Problem Variables.- 4.3 The Helmholtz Equation.- 4.3.1 DRM Formulations.- 4.3.2 DRM Results for Vibrating Beam.- 4.3.3 Results for Non-Inversion DRM.- 4.4 Non-Linear Cases.- 4.4.1 Burger's Equation.- 4.4.2 Spontaneous Ignition: The Steady-State Case.- 4.4.3 Non-Linear Material Problems.- 4.5 Computer Program 3.- 4.5.1 MAINP3.- 4.5.2 Subroutine ALFAF3.- 4.5.3 Subroutine RHSMAT.- 4.5.4 Subroutine DERIVXY.- 4.5.5 Results of Test Problems.- 4.6 Three-Dimensional Analysis.- 4.6.1 Equations of the Type ?2u = b(x, y, z).- 4.6.2 Equations of the Type ?2u = b(x, y, z, u).- 4.7 References.- 5 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u, t).- 5.1 Introduction.- 5.2 The Diffusion Equation.- 5.3 Computer Program 4.- 5.3.1 MAINP4.- 5.3.2 Subroutine ASSEMB.- 5.3.3 Subroutine VECTIN.- 5.3.4 Subroutine BOUNDC.- 5.3.5 Results of a Test Problem.- 5.3.6 Data Input.- 5.3.7 Computer Output.- 5.3.8 Further Applications.- 5.3.9 Other Time-Stepping Schemes.- 5.4 Special f Expansions.- 5.4.1 Axisymmetric Diffusion.- 5.4.2 Infinite Regions.- 5.5 The Wave Equation.- 5.5.1 Infinite and Semi-Infinite Regions.- 5.6 The Transient Convection-Diffusion Equation.- 5.7 Non-Linear Problems.- 5.7.1 Non-Linear Materials.- 5.7.2 Non-Linear Boundary Conditions.- 5.7.3 Spontaneous Ignition: Transient Case.- 5.8 References.- 6 Other Fundamental Solutions.- 6.1 Introduction.- 6.2 Two-Dimensional Elasticity.- 6.2.1 Static Analysis.- 6.2.2 Treatment of Body Forces.- 6.2.3 Dynamic Analysis.- 6.3 Plate Bending.- 6.4 Three-Dimensional Elasticity.- 6.4.1 Computational Formulation.- 6.4.2 Gravitational Load.- 6.4.3 Centrifugal Load.- 6.4.4 Thermal Load.- 6.5 Transient Convection-Diffusion.- 6.6 References.- 7 Conclusions.- Appendix 1.- Appendix 2.- The Authors.