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Progress on the Development of B-spline Collocation for the Solution of Differential Model Equations: A Novel Algorithm for Adaptive Knot Insertion
The application of collocation methods using spline basis functions to solve differential model equations has been in use for a few decades. However, the application of spline collocation to the solution of the nonlinear, coupled, partial differential equations (in primitive variables) that define the motion of fluids has only recently received much attention. The issues that affect the effectiveness and accuracy of B-spline collocation for solving differential equations include which points to use for collocation, what degree B-spline to use and what level of continuity to maintain. Success using higher degree B-spline curves having higher continuity at the knots, as opposed to more traditional approaches using orthogonal collocation, have recently been investigated along with collocation at the Greville points for linear (1D) and rectangular (2D) geometries. The development of automatic knot insertion techniques to provide sufficient accuracy for B-spline collocation has been underway. The present article reviews recent progress for the application of B-spline collocation to fluid motion equations as well as new work in developing a novel adaptive knot insertion algorithm for a 1D convection-diffusion model equation
Fast Isogeometric Boundary Element Method based on Independent Field Approximation
An isogeometric boundary element method for problems in elasticity is
presented, which is based on an independent approximation for the geometry,
traction and displacement field. This enables a flexible choice of refinement
strategies, permits an efficient evaluation of geometry related information, a
mixed collocation scheme which deals with discontinuous tractions along
non-smooth boundaries and a significant reduction of the right hand side of the
system of equations for common boundary conditions. All these benefits are
achieved without any loss of accuracy compared to conventional isogeometric
formulations. The system matrices are approximated by means of hierarchical
matrices to reduce the computational complexity for large scale analysis. For
the required geometrical bisection of the domain, a strategy for the evaluation
of bounding boxes containing the supports of NURBS basis functions is
presented. The versatility and accuracy of the proposed methodology is
demonstrated by convergence studies showing optimal rates and real world
examples in two and three dimensions.Comment: 32 pages, 27 figure
Boundary-Conforming Finite Element Methods for Twin-Screw Extruders using Spline-Based Parameterization Techniques
This paper presents a novel spline-based meshing technique that allows for
usage of boundary-conforming meshes for unsteady flow and temperature
simulations in co-rotating twin-screw extruders. Spline-based descriptions of
arbitrary screw geometries are generated using Elliptic Grid Generation. They
are evaluated in a number of discrete points to yield a coarse classical mesh.
The use of a special control mapping allows to fine-tune properties of the
coarse mesh like orthogonality at the boundaries. The coarse mesh is used as a
'scaffolding' to generate a boundary-conforming mesh out of a fine background
mesh at run-time. Storing only a coarse mesh makes the method cheap in terms of
memory storage. Additionally, the adaptation at run-time is extremely cheap
compared to computing the flow solution. Furthermore, this method circumvents
the need for expensive re-meshing and projections of solutions making it
efficient and accurate. It is incorporated into a space-time finite element
framework. We present time-dependent test cases of non-Newtonian fluids in 2D
and 3D for complex screw designs. They demonstrate the potential of the method
also for arbitrarily complex industrial applications
A novel envelope simulation technique for high-frequency nonlinear circuits
The paper proposes a new approach for the analysis and simulation of circuits subject to input signals with widely separated rates of variation. Such signals arise in communication circuits when an RF carrier is modulated by a low-frequency information signal. The approach will involve converting the ordinary differential equation system that describes the circuit to a partial differential equation system and subsequently solving the resultant system using a multiresolution collocation approach involving a cubic spline wavelet-based decomposition
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