Metric based up-scaling

We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only H\"{o}lder continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.Comment: Final version. Accepted for publication in Communications on Pure and Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April), Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution figures are available at http://www.acm.caltech.edu/~owhadi

Discrete Differential Geometry

Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session

Dynamic remeshing and applications

Triangle meshes are a flexible and generally accepted boundary representation for complex geometric shapes. In addition to their geometric qualities such as for instance smoothness, feature sensitivity ,or topological simplicity, intrinsic qualities such as the shape of the triangles, their distribution on the surface and the connectivity is essential for many algorithms working on them. In this thesis we present a flexible and efficient remeshing framework that improves these "intrinsic\u27; properties while keeping the mesh geometrically close to the original surface. We use a particle system approach and combine it with an iterative remeshing process in order to trim the mesh towards the requirements imposed by different applications. The particle system approach distributes the vertices on the mesh with respect to a user-defined scalar-field, whereas the iterative remeshing is done by means of "Dynamic Meshes\u27;, a combination of local topological operators that lead to a good natured connectivity. A dynamic skeleton ensures that our approach is able to preserve surface features, which are particularly important for the visual quality of the mesh. None of the algorithms requires a global parameterization or patch layouting in a preprocessing step, but works with simple local parameterizations instead. In the second part of this work we will show how to apply this remeshing framework in several applications scenarios. In particular we will elaborate on interactive remeshing, dynamic, interactive multiresolution modeling, semiregular remeshing and mesh simplification and we will show how the users can adapt the involved algorithms in a way that the resulting mesh meets their personal requirements

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Efficient CAD based adjoint optimization of turbomachinery using adaptive shape parameterization

The present thesis incorporates the CAD model into an adjoint-based optimization loop and uses it for the shape optimization of a 2D transonic turbine blade mid-section (profile). This is demonstrated by performing a single and multipoint optimization of the LS89 turbine, originally designed at the VKI. Substantial aerodynamic improvements are reported for both design point and off-design conditions.The case is deeply analysed from the flow analysis point of view. The present thesis is a step forward in three main aspects. First, the way the CAD model (for turbomachinery applications) is used within the shape optimization loop.To include the CAD model into the optimization loop, the CAD kernel and the grid generator (multiblock structured) are differentiated using the Algorithmic Differentiation (AD) tool ADOL-C. The advantage of including the CAD model in the design system is that assembly or manufacturing constraints can be imposed on the shape, allowing the optimized model or component to be manufactured. Second, a new definition of the parametric effectiveness indicator is proposed, based on the ability of a set of CAD-based design variables to produce a shape change using the adjoint sensitivities. An interesting thing is that parametric effectiveness considers the design variables can be non-orthogonal to each other and it can be applied to any type of constrained or unconstrained problems. If, in the beginning of the optimization, the parametric effectiveness is high, it is expected to reach a final solution with increased performance. Third, a new adaptive shape parameterization strategy is adopted, which is assisted by the above parametric effectiveness indicator in order to explore the design space more efficiently. The parametric effectiveness, which rates the quality of a CAD based parameterization for optimization, is used in a novel multilevel shape refinement procedure to: (1) introduce the minimum amount of design variables required to modify the shape in the direction the adjoint sensitivities dictate; (2) to create the best parameterization to be used during the optimization. By using the proposed methods and tools, not only the optimal geometry is defined by the CAD, which is the industry adopted standard for the design of components, but also, the designer avoids the use of either too few (slow improvements from cycle to cycle) or too many (increase the computational burden) design variables. The proposed methodology results to be an effective strategy to explore rich design spaces, to improve convergence rate, robustness and final solution of the adjoint-based optimization.Aquesta tesi incorpora el model de CAD en un procés iteratiu d'optimització basat en el mètode adjunt i l'utilitza per a l'optimització de la secció d'una turbina transónica 2D (perfil). Això es demostra realitzant una optimització de punt únic i multipunt de la turbina LS89, originalment dissenyada en el VKI. Es reporten millores aerodinàmiques substancials tant per al punt de disseny com per les condicions fora del disseny. El cas s'analitza en profunditat des del punt de vista aerodinàmic. Aquesta tesi representa un avanç en tres aspectes principals. Primer, la forma en què es fa servir el model CAD (per a aplicacions de turbomàquines) dins el procés d'optimització. Per incloure el model CAD en el bucle d'optimització, s'apliquen tècniques de diferenciació algorítmica (l'eina ADOL-C) al kernel del CAD i el generador de la malla (estructurada i multibloc). L'avantatge d'incloure el model CAD en el sistema de disseny és que es poden imposar restriccions de fabricació a la geometria, i això permet que el disseny ja optimitzat es pugui fabricar. En segon lloc, es proposa una nova definició de l'indicador d'efectivitat paramètrica, basat en la capacitat de produir el canvi en la geometria que dicta el mètode adjunt mitjançant l'ús de les variables de disseny que defineixen el model CAD. Cal destacar que l'efectivitat paramètrica considera que les variables de disseny poden ser no ortogonals entre si i es pot aplicar a qualsevol tipus de problemes restringits o no restringits. Si, al començament de l'optimització, l'efectivitat paramètrica és alta, s'espera que l'optimització arribi a una solució final amb major rendiment. En tercer lloc, s'adopta una nova estratègia de parametrització adaptativa, que és assistida per l'indicador d'efectivitat paramètrica anterior per explorar l'espai de disseny de manera més eficient. L'efectivitat paramètrica, que classifica la qualitat d'una parametrització basada en CAD per a l'optimització, s'utilitza en un nou procediment de refinament multinivell per: (1) introduir la quantitat mínima de variables de disseny requerides per modificar la geometria en la direcció que dicten les sensibilitats del mètode adjunt; (2) per crear la millor parametrització que s'utilitzarà durant l'optimització. En utilitzar els mètodes i eines proposats, no només la geometria òptima està definida en el model CAD, que és l'estàndard adoptat per la indústria per al disseny de components, sinó que també el dissenyador evita l'ús de molt poques (millores lentes de cicle a cicle) o massa variables de disseny (augmenten la càrrega computacional). La metodologia proposada resulta ser una estratègia efectiva per explorar espais de disseny enriquits, millora la taxa de convergència, la solidesa i la solució final de l'optimització basada en el mètode adjunt

Efficient CAD based adjoint optimization of turbomachinery using adaptive shape parameterization

The present thesis incorporates the CAD model into an adjoint-based optimization loop and uses it for the shape optimization of a 2D transonic turbine blade mid-section (profile). This is demonstrated by performing a single and multipoint optimization of the LS89 turbine, originally designed at the VKI. Substantial aerodynamic improvements are reported for both design point and off-design conditions.The case is deeply analysed from the flow analysis point of view. The present thesis is a step forward in three main aspects. First, the way the CAD model (for turbomachinery applications) is used within the shape optimization loop.To include the CAD model into the optimization loop, the CAD kernel and the grid generator (multiblock structured) are differentiated using the Algorithmic Differentiation (AD) tool ADOL-C. The advantage of including the CAD model in the design system is that assembly or manufacturing constraints can be imposed on the shape, allowing the optimized model or component to be manufactured. Second, a new definition of the parametric effectiveness indicator is proposed, based on the ability of a set of CAD-based design variables to produce a shape change using the adjoint sensitivities. An interesting thing is that parametric effectiveness considers the design variables can be non-orthogonal to each other and it can be applied to any type of constrained or unconstrained problems. If, in the beginning of the optimization, the parametric effectiveness is high, it is expected to reach a final solution with increased performance. Third, a new adaptive shape parameterization strategy is adopted, which is assisted by the above parametric effectiveness indicator in order to explore the design space more efficiently. The parametric effectiveness, which rates the quality of a CAD based parameterization for optimization, is used in a novel multilevel shape refinement procedure to: (1) introduce the minimum amount of design variables required to modify the shape in the direction the adjoint sensitivities dictate; (2) to create the best parameterization to be used during the optimization. By using the proposed methods and tools, not only the optimal geometry is defined by the CAD, which is the industry adopted standard for the design of components, but also, the designer avoids the use of either too few (slow improvements from cycle to cycle) or too many (increase the computational burden) design variables. The proposed methodology results to be an effective strategy to explore rich design spaces, to improve convergence rate, robustness and final solution of the adjoint-based optimization.Aquesta tesi incorpora el model de CAD en un procés iteratiu d'optimització basat en el mètode adjunt i l'utilitza per a l'optimització de la secció d'una turbina transónica 2D (perfil). Això es demostra realitzant una optimització de punt únic i multipunt de la turbina LS89, originalment dissenyada en el VKI. Es reporten millores aerodinàmiques substancials tant per al punt de disseny com per les condicions fora del disseny. El cas s'analitza en profunditat des del punt de vista aerodinàmic. Aquesta tesi representa un avanç en tres aspectes principals. Primer, la forma en què es fa servir el model CAD (per a aplicacions de turbomàquines) dins el procés d'optimització. Per incloure el model CAD en el bucle d'optimització, s'apliquen tècniques de diferenciació algorítmica (l'eina ADOL-C) al kernel del CAD i el generador de la malla (estructurada i multibloc). L'avantatge d'incloure el model CAD en el sistema de disseny és que es poden imposar restriccions de fabricació a la geometria, i això permet que el disseny ja optimitzat es pugui fabricar. En segon lloc, es proposa una nova definició de l'indicador d'efectivitat paramètrica, basat en la capacitat de produir el canvi en la geometria que dicta el mètode adjunt mitjançant l'ús de les variables de disseny que defineixen el model CAD. Cal destacar que l'efectivitat paramètrica considera que les variables de disseny poden ser no ortogonals entre si i es pot aplicar a qualsevol tipus de problemes restringits o no restringits. Si, al començament de l'optimització, l'efectivitat paramètrica és alta, s'espera que l'optimització arribi a una solució final amb major rendiment. En tercer lloc, s'adopta una nova estratègia de parametrització adaptativa, que és assistida per l'indicador d'efectivitat paramètrica anterior per explorar l'espai de disseny de manera més eficient. L'efectivitat paramètrica, que classifica la qualitat d'una parametrització basada en CAD per a l'optimització, s'utilitza en un nou procediment de refinament multinivell per: (1) introduir la quantitat mínima de variables de disseny requerides per modificar la geometria en la direcció que dicten les sensibilitats del mètode adjunt; (2) per crear la millor parametrització que s'utilitzarà durant l'optimització. En utilitzar els mètodes i eines proposats, no només la geometria òptima està definida en el model CAD, que és l'estàndard adoptat per la indústria per al disseny de components, sinó que també el dissenyador evita l'ús de molt poques (millores lentes de cicle a cicle) o massa variables de disseny (augmenten la càrrega computacional). La metodologia proposada resulta ser una estratègia efectiva per explorar espais de disseny enriquits, millora la taxa de convergència, la solidesa i la solució final de l'optimització basada en el mètode adjunt.Postprint (published version