# Parametric solutions involving geometry integrated with computer-aided design

## TL;DR

This paper introduces a general framework for computing parametric solutions in geometry using NURBS within CAD environments, enabling efficient sensitivity analysis and integration with engineering workflows.

## Contribution

It presents a novel, general approach for off-line parametric solutions involving NURBS geometries, applicable to 2D and 3D, with integration into CAD and efficient PGD-based computations.

## Key findings

- Valid for 2D and 3D geometries
- Achieves optimal convergence rates for flow problems
- Circumvents curse of dimensionality with PGD

## Abstract

The main objective of this work is to describe a general and original approach for computing an off-line solution for a set of parameters describing the geometry of the domain. That is, a solution able to include information for different geometrical parameter values and also allowing to compute readily the sensitivities. Instead of problem dependent approaches, a general framework is presented for standard engineering environments where the geometry is defined by means of NURBS. The parameters controlling the geometry are now the control points characterising the NURBS curves or surfaces. The approach proposed here, valid for 2D and 3D scenarios, allows a seamless integration with CAD preprocessors. The proper generalised decomposition (PGD), which is applied here to compute explicit geometrically parametrised solutions, circumvents the curse of dimensionality. Moreover, optimal convergence rates are shown for PGD approximations of incompressible flows.