Novel Discretization Schemes for the Numerical Simulation of Membrane Dynamics

Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements, and the entire space-time domain was discretized and solved simultaneously. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate

Real-time Digital Simulation of Guitar Amplifiers as Audio Effects

Práce se zabývá číslicovou simulací kytarových zesilovačů, jakož to nelineárních analogových hudebních efektů, v reálném čase. Hlavním cílem práce je návrh algoritmů, které by umožnily simulaci složitých systémů v reálném čase. Tyto algoritmy jsou prevážně založeny na automatizované DK-metodě a aproximaci nelineárních funkcí. Kvalita navržených algoritmů je stanovana pomocí poslechových testů.The work deals with the real-time digital simulation of guitar amplifiers considered as nonlinear analog audio effects. The main aim is to design algorithms which are able to simulate complex systems in real-time. These algorithms are mainly based on the automated DK-method and the approximation of nonlinear functions. Quality of the designed algorithms is evaluated using listening tests.

Simulation of Harmonic Oscillators on the Lattice

[EN] This work deals with the simulation of a two¿dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second¿order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state¿of¿the¿art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two¿dimensional lattice and check its energy conservation.This work has been supported by the Spanish Ministerio de Economia y Competitividad, the European Regional Development Fund (ERDF) under grant TIN2017-89314-P, and the Programa de Apoyo a la Investigacion y Desarrollo 2018 (PAID-06-18) of the Universitat Politecnica de Valencia under grant SP20180016.Tung, MM.; Ibáñez González, JJ.; Defez Candel, E.; Sastre, J. (2020). Simulation of Harmonic Oscillators on the Lattice. Mathematical Methods in the Applied Sciences. 43(14):8237-8252. https://doi.org/10.1002/mma.6510S823782524314Dehghan, M., & Hajarian, M. (2009). Determination of a matrix function using the divided difference method of Newton and the interpolation technique of Hermite. Journal of Computational and Applied Mathematics, 231(1), 67-81. doi:10.1016/j.cam.2009.01.021Dehghan, M., & Hajarian, M. (2010). Computing matrix functions using mixed interpolation methods. Mathematical and Computer Modelling, 52(5-6), 826-836. doi:10.1016/j.mcm.2010.05.013Kazem, S., & Dehghan, M. (2017). Application of finite difference method of lines on the heat equation. Numerical Methods for Partial Differential Equations, 34(2), 626-660. doi:10.1002/num.22218Kazem, S., & Dehghan, M. (2018). Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL). Engineering with Computers, 35(1), 229-241. doi:10.1007/s00366-018-0595-5Paterson, M. S., & Stockmeyer, L. J. (1973). On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials. SIAM Journal on Computing, 2(1), 60-66. doi:10.1137/0202007Sastre, J., Ibáñez, J., Defez, E., & Ruiz, P. (2011). Efficient orthogonal matrix polynomial based method for computing matrix exponential. Applied Mathematics and Computation, 217(14), 6451-6463. doi:10.1016/j.amc.2011.01.004Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Sastre, J., Ibáñez, J., Defez, E., & Ruiz, P. (2011). Accurate matrix exponential computation to solve coupled differential models in engineering. Mathematical and Computer Modelling, 54(7-8), 1835-1840. doi:10.1016/j.mcm.2010.12.049Serbin, S. M., & Blalock, S. A. (1980). An Algorithm for Computing the Matrix Cosine. SIAM Journal on Scientific and Statistical Computing, 1(2), 198-204. doi:10.1137/0901013Ruiz, P., Sastre, J., Ibáñez, J., & Defez, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics, 291, 370-379. doi:10.1016/j.cam.2015.04.001Higham, N. J. (1988). FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation. ACM Transactions on Mathematical Software, 14(4), 381-396. doi:10.1145/50063.21438

Design of an essentially non-oscillatory reconstruction procedure in finite-element type meshes

An essentially non oscillatory reconstruction for functions defined on finite element type meshes is designed. Two related problems are studied: the interpolation of possibly unsmooth multivariate functions on arbitary meshes and the reconstruction of a function from its averages in the control volumes surrounding the nodes of the mesh. Concerning the first problem, the behavior of the highest coefficients of two polynomial interpolations of a function that may admit discontinuities of locally regular curves is studied: the Lagrange interpolation and an approximation such that the mean of the polynomial on any control volume is equal to that of the function to be approximated. This enables the best stencil for the approximation to be chosen. The choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, two methods were studied: one based on an adaptation of the so called reconstruction via deconvolution method to irregular meshes and one that lies on the approximation on the mean as defined above. The first method is conservative up to a quadrature formula and the second one is exactly conservative. The two methods have the expected order of accuracy, but the second one is much less expensive than the first one. Some numerical examples are given which demonstrate the efficiency of the reconstruction

A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation

Computer algebra and transputers applied to the finite element method

Recent developments in computing technology have opened new prospects for computationally intensive numerical methods such as the finite element method. More complex and refined problems can be solved, for example increased number and order of the elements improving accuracy. The power of Computer Algebra systems and parallel processing techniques is expected to bring significant improvement in such methods. The main objective of this work has been to assess the use of these techniques in the finite element method. The generation of interpolation functions and element matrices has been investigated using Computer Algebra. Symbolic expressions were obtained automatically and efficiently converted into FORTRAN routines. Shape functions based on Lagrange polynomials and mapping functions for infinite elements were considered. One and two dimensional element matrices for bending problems based on Hermite polynomials were also derived. Parallel solvers for systems of linear equations have been developed since such systems often arise in numerical methods. Both symmetric and asymmetric solvers have been considered. The implementation was on Transputer-based machines. The speed-ups obtained are good. An analysis by finite element method of a free surface flow over a spillway has been carried out. Computer Algebra was used to derive the integrand of the element matrices and their numerical evaluation was done in parallel on a Transputer-based machine. A graphical interface was developed to enable the visualisation of the free surface and the influence of the parameters. The speed- ups obtained were good. Convergence of the iterative solution method used was good for gated spillways. Some problems experienced with the non-gated spillways have lead to a discussion and tests of the potential factors of instability

Numerical Methods for Uncertainty Quantification in Gas Network Simulation

When modeling the gas flow through a network, some elements such as pressure control valves can cause kinks in the solution. In this thesis we modify the method of simplex stochastic collocation such that it is applicable to functions with kinks. First, we derive a system of partial differential and algebraic equations describing the gas flow through different elements of a network. Restricting the gas flow to an isothermal and stationary one, the solution can be determined analytically. After introducing some common methods for the forward propagation of uncertainty, we present the method of simplex stochastic collocation to approximate functions of uncertain parameters. By utilizing the information whether a pressure regulator is active or not in the current simulation, we improve the method such that we can prove algebraic convergence rates for functions with kinks. Moreover, we derive two new error estimators for an adaptive refinement and show that multiple refinements are possible. Conclusively, several numerical results for a real gas network are presented and compared with standard methods to demonstrate the significantly better convergence results