This work concerns the development of a meshfree semi-implicit numerical scheme based on the Smoothed Particle Hydrodynamics (SPH) method, here applied to free surface hydrodynamic problems governed by the shallow wat...
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ISBN:
(数字)9783319519548
ISBN:
(纸本)9783319519548;9783319519531
This work concerns the development of a meshfree semi-implicit numerical scheme based on the Smoothed Particle Hydrodynamics (SPH) method, here applied to free surface hydrodynamic problems governed by the shallow water equations. In explicit numerical methods, a severe limitation on the time step is often due to stability restrictions imposed by the CFL condition. In contrast to this, we propose a semi-implicit SPH scheme, which leads to an unconditionally stable method. To this end, the discrete momentum equation is substituted into the discrete continuity equation to obtain a linear system of equations for only one scalar unknown, the free surface elevation. The resulting system is not only sparse but moreover symmetric positive definite. We solve this linear system by a matrix-free conjugate gradient method. Once the new free surface location is known, the velocity can directly be computed at the next time step and, moreover, the particle positions can subsequently be updated. The resulting meshfree semi-implicit SPH method is validated by using a standard model problem for the shallow water equations.
When a symmetric block diagonal matrix [GRAPHICS] undergoes an [off-diagonal perturbation [GRAPHICS] the eigenvalues of these matrices are known to differ only by O(parallel to E-12 parallel to(2)/gap), which scales q...
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ISBN:
(纸本)9783319624266;9783319624242
When a symmetric block diagonal matrix [GRAPHICS] undergoes an [off-diagonal perturbation [GRAPHICS] the eigenvalues of these matrices are known to differ only by O(parallel to E-12 parallel to(2)/gap), which scales quadratically with the norm of the perturbation. Here gap measures the distance between eigenvalues, and plays a key role in the constant. Closely related is the first-order perturbation expansion for simple eigenvalues of a matrix. It turns out that the accuracy of the first-order approximation is also O(parallel to E parallel to(2)/gap), where E is the perturbation matrix. Also connected is the residual bounds of approximate eigenvalues obtained by the Rayleigh-Ritz process, whose accuracy again scales quadratically in the residual, and inverse-proportionally with the gap between eigenvalues. All these are tightly linked, but the connection appears to be rarely discussed. This work elucidates this connection by showing that all these results can be understood in a unifying manner via the quadratic perturbation bounds of block diagonal matrices undergoing off-diagonal perturbation. These results are essentially known for a wide range of eigenvalue problems: symmetric eigenproblems (for which the explicit constant can be derived), nonsymmetric and generalized eigenvalue problems. We also extend such results to matrix polynomials, and show that the accuracy of a first-order expansion also scales as O(parallel to E parallel to(2)/gap), and argue that two-sided projection methods are to be preferred to one-sided projection for nonsymmetric eigenproblems, to obtain higher accuracy in the computed eigenvalues.
Numerical methods for singularly perturbed convection-diffusion problems posed on annular domains are constructed and their performance is examined for a range of small values of the singular perturbation parameter. A...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
Numerical methods for singularly perturbed convection-diffusion problems posed on annular domains are constructed and their performance is examined for a range of small values of the singular perturbation parameter. A standard polar coordinate transformation leads to a transformed elliptic operator containing no mixed second order derivative and the transformed problem is then posed on a rectangular domain. In the radial direction, a piecewise-uniform Shishkin mesh is used. This mesh captures any boundary layer appearing near the outflow boundary. The performance of such a method is examined in the presence or absence of compatibility constraints at characteristic points, which are associated with the reduced problem.
In domain decomposition we decompose a global domain into subdomains. The shape and form of these subdomains may have an effect upon the overall performance of our methods. In this paper we study the effects of irregu...
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The paper describes numerical prediction of aerodynamic noise generated from the aircraft. It focuses on the simulation of turbulent flow around rectified flap on the wing represented in 2D. Simulation of turbulent fl...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
The paper describes numerical prediction of aerodynamic noise generated from the aircraft. It focuses on the simulation of turbulent flow around rectified flap on the wing represented in 2D. Simulation of turbulent flow is modeled using the stabilized orthogonal subgrid scale (OSGS) method with dynamical subscales. It is shown how the stabilization method can perform simulation of turbulent flow affecting the prediction of acoustic sources calculated applying Lighthill's analogy. Acoustic sources are used in inhomogeneous Helmholtz equation to simulate pressure wave propagation in the domain closing the circle of three main steps required for simulating aeroacoustics phenomena. It is shown that OSGS with dynamical subscales gives better representation of the spectrum. Overall, better prediction of energy transfer across large and small eddies provides better allocation and presentation of acoustics sources. These sources change wave propagation of the pressure in acoustic field.
This paper takes a new look at regression with adaptive sparse grids. Considering sparse grid refinement as an optimisation problem, we show that it is in fact an instance of submodular optimisation with a cardinality...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
This paper takes a new look at regression with adaptive sparse grids. Considering sparse grid refinement as an optimisation problem, we show that it is in fact an instance of submodular optimisation with a cardinality constraint. Hence, we are able to directly apply results obtained in combinatorial optimisation research concerned with submodular optimisation to the grid refinement problem. Based on these results, we derive an efficient refinement indicator that allows the selection of new grid indices with finer granularity than was previously possible. We then implement the resulting new refinement procedure using an averaged stochastic gradient descent method commonly used in online learning methods. As a result we obtain a new method for training adaptive sparse grid models. We show both for synthetic and real-life data that the resulting models exhibit lower complexity and higher predictive power compared to currently used state-of-the-art methods.
We present the dual reciprocity boundary element method (DRBEM) solution of the system of equations which model magnetohydrodynamic (MHD) flow in a pipe with moving lid at low magnetic Reynolds number. The external ma...
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ISBN:
(纸本)9783319399294;9783319399270
We present the dual reciprocity boundary element method (DRBEM) solution of the system of equations which model magnetohydrodynamic (MHD) flow in a pipe with moving lid at low magnetic Reynolds number. The external magnetic field acts in the pipe-axis direction generating the electric potential. The solution is obtained in terms of stream function, vorticity and electric potential in the cross-section of the pipe, and the pipe axis velocity is also computed under a constant pressure gradient. It is found that fluid flow concentrates through the upper right corner forming boundary layers with the effect of moving lid and increased magnetic field intensity. Electric field behavior is changed accordingly with the insulated and conducting portions of the pipe walls. Fluid moves in the pipe-axis direction with an increasing rate of magnitude when Hartmann number increases. The boundary only nature of DRBEM provides the solution at a low computational expense.
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