The simulation of wave phenomena in heterogeneous media has been a very active field of research. Depending on the specific problem that we are interested in solving, different techniques are here introduced. Firstly, Hybrid DG methods (or HDG) are developed for these kind of problems. These methods are fully implicit, unstructured, and high-order accurate in both space and time; yet they are computationally attractive because the only globally coupled unknown is the numerical trace of the velocity field. Since the numerical trace is only defined on the element faces and single-valued, the HDG methods may have significantly less global degrees of freedom than other DG methods using implicit time integration. Another attractive feature of the HDG methods is that they yield optimal convergence. The figures below show wave fields obtained using this numerical methodology.
1d Bandgap problem Planar wave interaction with an object on a surface and a cylinder
However, some of the problems that we are particularly interested in solving have geometric details and solution patterns that belong to very different scales of size. Moreover, repetition of patterns is often very common in these problems; even infinite repetition when periodic. The approach that is here introduced is called a MultiScale CG method, or MSCG and it defines the subproblems at a subdomain level, in contrast to classical HDG or HCG. This subtle extension is crucial because it provides further reduction in the global degrees of freedom and better efficiency in parallelization. Moreover, it renders this approach very attractive for problems with repeated, piecewise-constant, or periodic coefficients, since in such cases the number of local subproblems can be significantly reduced by a judicious choice of subdomains and associated approximation spaces. With these technology, problems like the ones shown in the figures below can be accurately solved with an extremely high efficiency. The methodology has been developed for 2d and 3d applications.
Superlensing 2d (above) - 3d (below) photonic waveguide Two dimensional Photonic Sharp Bend
Discrete Design Optimization of Structured Materials for wave problems
The design of materials is often based on physical and mathematical intuition. However, when finite domains are considered and manufacturability concerns want to be addressed, they might not be enough. Recently, several optimization techniques have successfully been extended and applied for design problems. Semidefinite or convex optimization techniques have already been used for the design of materials, as well as Nonlinear programming tools and Adjoint methods. Nevertheless, all these methods do not impose the discrete nature of the design variables and very often lead to continuous distributions of materials. In fact, what we seek is only using a finite pool of materials that are available. To that end, a local search discrete optimization procedure has been developed. Some results can be seen in the figures below.
Solution and scattered fields of the cloaking problem. Without cloak and with the optimal distribution.
Photonic Bandgap Optimization
Photonic crystals have been a topic of high interest within the last few years, specially since in 1987 the 2d bandgap in periodic structures were found and its existence was proved. The symmetries that the crystal inherits from its periodicity give it some unique properties. Maximizing the gap opened between some two consecutive bands in the dispersion for both the TE and the TM polarization at the same time warrantees that for the frequencies given in the gap, there will be no k in which it will propagate; so it will exponentially decay.
It is then crucial to analyze the properties of such symmetries from a group theory perspective. A very interesting property that photonic crystals have comes from the Bloch's theorem if there is translational symmetry in a given lattice. With all, periodic structures exhibiting electromagnetic wave band gaps, or photonic crystals, have proven very important applications as device components for integrated optics including frequency filters, waveguides [see figure], switches, and optical buffers.
Shape optimization is used in order to get the unit lattice solution and bandgaps of over 40% (in terms of gap-midgap ratio) has been found for some square and triangular lattices. An initial guess must be given which is evolved iteratively until it gets into an optimal solution. The figure in the left shows one optimized structure with its band diagram representation and a gap-midgap ratio of over 30%.
Most of this work is part of the research worked out by Abby Men for her Ph.D Thesis together with the advise of Professors Robert Freund
, Pablo Parrilo
and Jaume Peraire
Consider for instance a one dimensional beam under heat transfer phenomena. It might be useful to maximize its inner structure such that the transferred heat, at a given time and for a given amount of material, is maxima, or minima. That is, we seek to minimize (or maximize) the time that it will take to the whole beam to get the prescribed temperature given as a source in the two extrema of the beam which is the same as maximizing (or minimizing) the heat transfer by the sources into the beam.
This kind of problems are governed by a system of ODE's which can be discretized using the finite element method. In the end, it turns out to be a convex optimization problem which gives a continuous distribution of material. However, it turns out that it would even be far more useful to get a discrete solution in which we can choose between one or another material for each discrete element. The discrete problem is, nevertheless, a nonconvex optimization problem, NP hard actually.
Using a subspace approximation one can obtain, always after choosing a good enough starting solution, the optimal discrete distribution of the two materials that optimizes the heat transfer. The attached figures show how the solutions are and how better the solution is from the naive constant solution.
Prof. A. Huerta
Prof. J. Peraire
Prof. J. Bonet
The aim of the project is to obtain a new methodology for the calculation of the collapse state in structures. It is based on the lower bound theorem of the limit analysis and solved by optimization techniques. The lower bound theorem asserts that possible solution states fulfils equilibrium and do not violate the yield constraints. In this work, the collapsed state is found by means of searching the generalized stress distribution, subjected to the equilibrium and yield constraints, that maximizes the external loading multiplier. Apart from the general procedure based in generalized stresses, the main novelty lies on the adaptive linearisation of the yield quadratic forms as well as the formulation for the plastic flow in the space of bending moments and axial forces. Also, in order to obtain nonnegative variables a new strategy that avoids to introduce too many variables is implemented. This technique involves just one additional variable.
Regarding this improvement of teaching, we have also filmed some lectures and uploaded them to the Moodle platform webpage and we have even written a book in which we put all this content available, not only for our course students but also for everyone who want to reach them, both instructors and students.
Let's also say that all the programmed contents are done with the mathematical software Wiris
, which is specially great to represent figures and it has been included in the Moodle platform in order to be able to create this contents directly on Moodle without attempting to use html
, so it couldn't be more userfriendly. That's what instructors like most!
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