**The main scientific challenge in the MOTOR project is to develop novel CAE technologies that enable the accurate and unified representation of fluid energy machine geometries throughout all stages of the design, simulation, and optimization chain. This is combined with the challenge to combine the novel technologies with existing CAD and CAE tools and to integrate them into design workflows in industrial environments. To achieve these ambitious goals the MOTOR project relies on several key technologies.**

Freeform surfaces are widely used in CAD systems to describe the skin of three-dimensional geometric objects such as turbine blades, car bodies and boat hulls. Most CAD systems nowadays adopt non-uniform rational B-splines (NURBS) as sound mathematical tool to describe the surface forms. NURBS can represent virtually any desired shape and allow for a flexible control over the surface via the position of the control points. In essence, a surface patch is defined by its corner points and the intermediate control points act like magnets that attract the surface giving it the final shape.

More complex surfaces are constructed by gluing several patches together ensuring that the connection of surfaces is sufficiently smooth (continuous or even continuously differentiable).

The direct construction of freeform surfaces by moving control points around is, however, not that useful since fluid energy machines are typically defined in terms of design space variables such as thickness distributions and pitch angles from which the CAD model is generated automatically by application-specific pre-processing tools.

To enable numerical simulations of the mechanical behaviour of the geometry object and the flow around it the parametric description of the geometry surface needs to be extended into an analysis-suitable computational mesh or, in the case of Isogeometric Analysis, multi-patch volumetric parameterizations.

The MOTOR project develops

- A
**geometry-aware volumetric mesh generator**that creates tetrahedral/hexahedral and mixed tet/hex volumetric meshes for multi-domain geometries that match at the interfaces between two disciplines and represent curved boundaries and interfaces accurately to enable multi-physics simulations with enhanced accuracy. It builds on the commercial pre-processing tool MERGE which provides automatic repairing of erroneous meshes (cleaning of non-manifold meshes, removal of disturbances, closing of holes) and merging of multiple input objects together. - A
**geometry-aware block structuring and volumetric parameterization**pipeline that automatically segments three-dimensional free-form domains into topological hexahedra (blocks) and creates NURBS volume parameterizations suitable for isogeometric numerical simulations

The MOTOR project adopts advanced spline technologies such as truncated hierarchical (TH)B-splines [1,2] and extends them to the fully multi-variate case to support adaptive refinement in 3D solids.

Next to the modelling of the fluid energy machine geometries statically, shape deformation techniques are one of the fundamental ingredients for the automated optimization of mechanical components. Standard deformation techniques typically act on the set of discrete grid points of the computational mesh, and therefore, lack a direct link to the parameterized description of the CAD geometry not to speak of its creating design space. This requires an extra post-processing step to approximate the optimized grid by a CAD geometry, which is however not exact and introduces additional errors in the geometric quality. Moreover, imposing constraints like a minimal thickness distribution is a non-trivial task.

The MOTOR project investigates **advanced shape deformation techniques based on volumetric approaches** that provide full CAD-compatibility and a flexible and efficient design space to overcome the limitations of standard deformation methods. This is achieved by defining

template mappings for certain types of geometries (blade of an aircraft engine or ship propeller, male/female rotor of screw machine) for which volumetric parameterizations and deformation can be generated a priori. This approach allows for integrating computational meshes, isogeometric segmentations in the deformation and enables hot-cold transformation between the manufactured (cold) geometry and its shape under operating conditions.

Bibliography

[1] C. Giannelli, B. Jüttler and H. Speleers: THB-splines: The truncated basis for hierarchical splines. Computer Aided Geometric Design, 2012:29, 485-498, doi: 10.1016/j.cagd.2012.03.025

[2] C. Giannelli, B. Jüttler, H. Speleers: Strongly stable bases for adaptively refined multilevel spline spaces. Advances in Computational Mathematics, Springer, 2014, 40:459-490, doi: 10.1007/s10444-013-9315-2

The MOTOR project develops software modules for CFD, CSM and FSI simulations building upon the state-of-the-art in Isogeometric Analysis. The design principle of this approach is to utilize the same mathematical formalism to represent the CAD geometry and the approximate solution to the fluid flow, structural mechanics of FSI problem.

The number of design parameters describing the main rotor geometries of fluid energy machines is typically very large. For instance, more than 50 design parameters are required to describe a single ship propeller. The curse of dimensionality practically prevents to fully explore optimization scenarios with such large degree of freedoms using gradient-free techniques such as those based on evolutionary algorithms. On the other hand, adjoint-based optimization methods bear the potential of being trapped in local optima leading to sub-optimal design shapes.

MOTOR tackles this issue by using a **hybrid multi-level shape optimization approach in the parameterization** so as to define the geometry in different levels of shape variety. A coarse model controls the geometry with only few design parameters, therefore, limiting the shape complexity. A hierarchy of more and more fine-grained models allows larger shape variability at the expense of a larger number of degrees of freedom. Geometry shapes can be exchanged between the different levels through approximation or shape refinement. This procedure allows for combining efficient design space exploration based on gradient-free methods operating on the coarse models with more local searches via adjoint-based optimization adopting the fine-grained model. The representation of the geometry by NURBS or THB-splines is a key ingredient for the hybridized approach making it possible to smoothly blend between coarse and fine geometry models without introducing approximation errors.

Most current adjoint-based optimization algorithms directly act on the discrete set of grid points of the computational and thus break the link with the underlying CAD geometry and its creating design space. This has as major drawback the need for a CAD refitting of the optimal shape, which impairs optimality. The MOTOR project therefore adopts **adjoint-based optimization with respect to the design parameters controlling the CAD model**, hence keeping the link with CAD. This has the major advantage that geometrical constraints such as minimal thickness distributions can easily be implemented and that the optimized shape is directly available in CAD format.