Computational Mathematics (CM)
http://hdl.handle.net/20.500.11824/5
2023-04-25T04:47:21ZGeneralized plane offsets and rational parameterizations
http://hdl.handle.net/20.500.11824/1588
Generalized plane offsets and rational parameterizations
Rochera, D.
In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown.
2023-04-21T00:00:00ZThe role of the particle aspect ratio in the discharge of a narrow silo
http://hdl.handle.net/20.500.11824/1583
The role of the particle aspect ratio in the discharge of a narrow silo
Pongó, T.; Fan, B.; Hernández-Delfin, D.; Török, J.; Stannarius, R.; Hidalgo, R. C.; Börzsönyi, T.
The time evolution of silo discharge is investigated for different granular materials made of spherical or elongated grains in laboratory experiments and with discrete element model (DEM) calculations. For spherical grains, we confirm the widely known typical behavior with constant discharge rate (except for initial and final transients). For elongated particles with aspect ratios between 2 ⩽ L/d ⩽ 6.1, we find a peculiar flow rate increase for larger orifices before the end of the discharge process. While the flow field is practically homogeneous for spherical grains, it has strong gradients for elongated particles with a fast-flowing region in the middle of the silo surrounded by a stagnant zone. For large enough orifice sizes, the flow rate increase is connected with a suppression of the stagnant zone, resulting in an increase in both the packing fraction and flow velocity near the silo outlet within a certain parameter range.
2022-10-01T00:00:00ZExploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature
http://hdl.handle.net/20.500.11824/1581
Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature
Croci, M.; Muñoz-Matute, J.
In this article, we propose an algorithm for approximating the action of $\varphi$-functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the $\varphi$-functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.
2023-02-04T00:00:00ZCombining DPG in space with DPG time-marching scheme for the transient advection–reaction equation
http://hdl.handle.net/20.500.11824/1580
Combining DPG in space with DPG time-marching scheme for the transient advection–reaction equation
Muñoz-Matute, J.; Demkowicz, L.; Roberts, N. V.
In this article, we present a general methodology to combine the Discontinuous Petrov-Galerkin (DPG) method in space and time in the context of methods of lines for transient advection-reaction problems. We rst introduce a semidiscretization in space with a DPG method rede ning the ideas of optimal testing and practicality of the method in this context. Then, we apply the recently developed DPG-based time-marching scheme, which is of exponential-type, to the resulting system of Ordinary Differential Equations (ODEs). We also discuss how to e ciently compute the action of the exponential of the matrix coming from the space semidiscretization without assembling the full matrix. Finally, we verify the proposed method for 1D+time advection-reaction problems showing optimal convergence rates for smooth solutions and more stable results for linear conservation laws comparing to the classical exponential integrators.
2022-12-01T00:00:00Z