# MATHDES

## Mathematical Design, Modelling and Simulations

Mathematical Design, Modeling, and Simulations (MATHDES) group works on the design, analysis, implementation, and optimization of numerical schemes for mathematical models arising from real-life applications.

*Principal Investigators:*

**Michael Barton**and

**David Pardo**

Our research spans areas of Deep Learning, Inverse Problems, Finite Elements, Massive Computations, Numerical Analysis, Geometric Modeling, Computer Aided Design, and Modeling of Manufacturing Processes. We work in close interaction with industrial partners and institutions to promote transfer of knowledge and obtain feedback from real-life applications.

We currently work on three European Projects:

(a) a FET-OPEN project on Analysis, Design, and Manufacturing using Microstructures ADAM^2, which aims at revolutionizing the design-analysis-manufacturing pipeline, coordinated by Michael Barton; (b) a Marie Curie RISE Action MATHROCKS, intended to improve and exchange interdisciplinary knowledge on applied mathematics, high performance computing, and geophysics, coordinated by David Pardo; and (c) the European NextGenerationEU project IA4TES, funded by the Spanish Ministry for Economic Affairs and Digital Transition and led by Iberdrola, for structural health monitoring and assessment of remaining useful life of offshore wind platforms.

We collaborate with multiple prestigious institutions and industrial partners, including The University of Texas at Austin (USA), Stratasys LTD (Israel), INRIA (France), Hutchinson S.A. (France), Seoul National University (South Korea), Technion - Israel Institute of Technology (Israel), MacQuaire University (Australia), Ecole Polytechnique Federale de Lausanne (Switzerland), Curtin University (Australia), Technical University of Wein (Austria), King Abdullah University of Science and Technology (Saudi Arabia), Software Competence Center Hagenberg (Austria), Pontifical Catholic University of Valparaíso (Chile), Pontifical Catholic University of Chile, National University of Colombia, Central University of Venezuela, University of Buenos Aires (Argentina), Barcelona Supercomputing Center (Spain), Polytechnic University of Catalunya (Spain), University of Barcelona (Spain), University Charles III of Madrid (Spain), Tecnalia (Spain), Trimek S.A. (Spain), and University of the Basque Country (UPV/EHU).

This BCAM group has strong collaboration ties with the sister UPV/EHU Group MATHMODE.

In particular, we are currently involved in the following research projects:

### Deep Learning Algorithms for Inverse Problems:

We study and develop Deep Learning methods to interpret (invert) in real-time elasto-acoustic and electromagnetic measurements. To achieve this objective, we design and implement proper Deep Neural Network architectures, loss functions, and error control algorithms enabling us to efficiently approximate physically meaningful inverse solutions.

### Massive Finite Element Simulations:

We develop a set of numerical Galerkin-based methods for producing massive synthetic datasets for training Deep Neural Networks. Some examples include variants of Isogeometric Analysis, Dimensionality Reduction Algorithms, Reduced Order Models, Proper Generalized Decompositions, and *hp*-Adaptivity.

### Geometric Modeling:

We model manufacturing processes such as 5-axis Computer Numerically Controlled (CNC) machining, see Fig.1, hot-wire cutting, or hybrid (additive and subtractive) manufacturing.

**Fig. 1**: 5-axis flank machinability of a reference surface (dark) using various machining tools (green). The tool-paths that meet fine machining tolerances are shown as ruled surface (motions of the tool axis, yellow).

### Manufacturing Applications:

In collaboration with the High Performance Manufacturing Group (UPV/EHU), we design path-planning algorithms for multi-axis CNC machining. An example of a simulation of 5-axis flank milling with a custom-shaped tool is shown in Fig. 2. The tool and its motion are both unknowns in our optimization-based framework. Manufactured result (blisk) machined with our algorithm using a conical tool is shown in Fig. 3.

**Fig. 2**: Simulation of path-planning for 5-axis double-flank CNC machining of a spiral bevel gear using a custom-shaped milling tool, (left). The 3D-printed prototype of the designed tool is shown in right.

**Fig. 3**: Finishing of the blisk blade using the state-of-the-art software (Siemens NX, left) and our algorithm (right). Additionally, to higher accuracy (not visible), observe the smooth light reflection in our result, that is due to negligible distance error between the neighboring milling paths.

### Geophysical applications:

We employ advanced numerical simulation and inversion methods to characterize the Earth´s subsurface, see Fig. 4. This is critical in several applications, such as (a) the prospecting of water, precious minerals, and hydrocarbons, (b) earthquake prediction and seismic hazard estimation, (c) seismic monitoring, (d) mine detection, (e) geothermal energy production, and (f) management of CO2 sequestration at the industrial scale that is needed to abate the global climate change problem.

**Fig. 4**: Controlled Source Electromagnetic Measurements for Earth Exploration

### Offshore Wind Energy application:

We design Deep Neural Network architectures as a support of the design and operation of emerging technologies for offshore wind energy applications in three areas: (a) for accelerating CFD simulations using OpenFoam for analysing the hydrodynamics of support structures; (b) for structural health monitoring of components (mooring systems, power cables) in offshore floating platforms (see Fig. 5).; (c) real-time control of offshore wind energy farms for Fault Detection and Fault Tolerant Control.

**Fig. 5**: Autoencoder architecture for Structural Health Monitoring of Floating offshore wind components

### Efficient Minimum Distance Computation for Solids of Revolution

Elber, G.; Kim, M.; Yoon, S.; Son, S. (2020-01-01)

We present a highly efficient algorithm for computing the minimum distance between two solids of revolution, each of which is defined by a planar cross-section region and a rotation axis. The boundary profile curve for th...

### Surface-Surface-Intersection Computation using a Bounding Volume Hierarchy with Osculating Toroidal Patches in the Leaf Nodes

Park, Y.; Son, S.; Kim, M.; Elber, G. (2022-01-01)

We present an efficient and robust algorithm for computing the intersection curve of two freeform surfaces using a Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches. The covering of e...

### A Review of a B-spline based Volumetric Representation: Design, Analysis and Fabrication of Porous and/or Heterogeneous Geometries

(2023-01-01)

### Implicit Functionally Graded Conforming Microstructures

Elber, G.; Kim, M.; Hong, Q.Y. (2023-01-01)

The tensor product parametric representations are the most commonly used representation in geometric modeling. Yet, other representations have advantages in certain aspects, and in this work, we focus on employing implicit ...

Geometry and Tool Motion Planning for Curvature Adapted CNC Machining

**Michal Bizzarri, Oleksii Sliusarenko, Florian Rist, Helmut Pottmann, Michael Barton**

## CL file

Datasets (Zenodo) for the paper "Manufacturing of screw rotors via 5-axis double-flank CNC machining"

**Authors: **Michael Barton, Michal Bizzarri

**License: **Zenodo

## CL file

Datasets (Zenodo) for the paper "Geometry and tool motion planning for curvature adapted CNC machining"

**Authors:** Michael Barton, Michal Bizzarri, Oleksii Sliusarenko, Florian Rist, Helmut Pottmann

**License:** Zenodo