### Mixed Dimensional Modelling

## Introduction

CAD is traditionally seen as the carrier of information in design. However, creating a finite element analysis model of a design invariably requires *simplification*. Therefore, the analysis model is often rebuilt from scratch, relying upon the judgment of skilled analysts, much of the work in creating the geometry being duplicated. As well as removing details from the design geometry, plate, shell and beam element types are also often used where they can give more accurate and cost effective results.

Procedures have been developed for the automatic dimensional reduction of a 2D geometric model to an equivalent 1D beam analysis model. This was achieved using the medial axis transform, an alternative skeleton-like representation of the geometric model having properties relevant to model simplification. Mixed dimensional models are created implicitly with the required coupling automatically applied.

Operations have also been defined and implemented for the expansion of these operations to the dimensional reduction of 3D solid models to equivalent solid/shell/beam analysis models. These operations are interactive, with appropriate physical properties such as shell thickness, beam section moment of areas and torsion constants calculated automatically.

The automatic dimensional reduction of thin walled solids to equivalent shell models has been implemented using a geometric measure similar to an aspect ratio, but extended for solid to shell dimensional reduction. This process uses the operations described earlier directly, with shell thicknesses being calculated automatically.

## Dimensional Reduction of 2D Geometric Models using the Medial Axis Transform

The first stage is to subdivide the 2D model into subregions which are suitable for representation by 1D beam elements, and those which are not to either by 2D elements, or by an equivalent 0D point mass.

A threshold value of aspect ratio can then be introduced, below which a region is not suitable for representation as 1D beam elements and must be represented either by the 2D region or an equivalent 0D point mass. All other regions can then be represented by 1D beams.

Progressive dimensional reduction based on aspect ratio

For the case of mixed 1D/0D abstracted models, the 1D beams can either be connected together with rigid links, or extrapolated to meet at their closest point inside the neighbouring 0D region.

## Analysis Results

The diagram below shows a 2D axisymmetric section of part of a turbine casing. They were used here to validate the dimensionally reduced beam models created. A natural frequency analysis of both the 2D sections and the abstracted beam models was carried out. The results for the first three modes are presented below. Three analyses were carried out, namely for a full 2D mesh, a 1D model with rigid link coupling and an extrapolated 1D beam model. Also shown is the extrapolated beam model (top left), the beams together with 2D regions represented as point masses in the analysis model (centre left) and the beam model coupled with rigid links (bottom left).

From the analysis results it can be seen that the frequencies obtained from the 2D models are bounded by the reduced model results. The frequencies for the rigid link model are higher, as might be expected since the full mass of the structure is represented, but some of the flexibility has been replaced by rigid links of infinite stiffness. The extrapolated beam models give lower natural frequency results than the full 2D models. In both cases the mode shapes obtained are representative of the 2D model, and the analysis results are good given the small number of degrees of freedom in the reduced models.

## Swept 3D Models

A lot of 2D planar models used are cross sections of a full 3D model which has shown some form of of symmetry. Therefore it was logical to extend the creation of analysis models to swept 2D shell / 1D beam stiffener models. These models are created automatically, with all required physical properties and coupling between element types applied. Some simple models are shown below.Section swept in Z-direction, coupled with rigid links and extrapolation

Section rotationally swept about x-axis, rigid links and extrapolation

## 2D Mixed Dimensional Models - Automatically Created

## Interactive Generation of Reduced Dimensional Models from 3D Solid Models

## Automatic Solid - Shell Dimensional Reduction

This model was dimensionally reduced to the equivalent shell model below automatically, no user interaction was required.

The thin walled solid model below was also dimensionally reduced automatically to a shell model, and meshed using triangular elements.