# Free-Form Patterning

### Patterning and Customization: Evaluating Tensor Field Generation For Mechanical Design On Free-Form Surfaces

###### Diego Fernando Andrade

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering

Carnegie Mellon University Pittsburgh, Pennsylvania May 2017

Computer Aided Design (CAD) systems currently offer tools for generating simple patterns, such as uniformly spaced rectangular or radial patterns. However, they are not applicable to more general cases required in industrial design, including arbitrarily shaped target geometry and graded feature sizes. These tools are limited in several ways: (1) They cannot be applied to free-form geometries used in industrial design, (2) Patterning of these features happens within a single working plane and is not applicable to highly curved surfaces, and (3) Created features lack anisotropy and spatial variations, such as changes in the size and orientation of geometric features within a given region.

This thesis proposes a new method of taking input for a target region along with sizing metrics. It will generate feature patterns automatically in three steps: (1) packing isotropic or anisotropic cells tightly in a target region, (2) scaling features according to the specified sizing metrics, and (3) adding features on the base geometry. This approach automatically generates complex patterns that conform to the boundary of any specified region.

User input of a small number of geometric features (called “seed features”) of desired size and orientation in preferred locations also can be specified within the target domain. These geometric seed features are then transformed into tensors and used as boundary conditions to generate a Riemannian metric tensor field. A form of the Laplace heat equation is used to generate the field over the target domain, subject to specified boundary conditions.

*Pix

The field represents the anisotropic pattern of the geometric features. The system is implemented as a plugin module in a commercial CAD package to add geometric features to the target region of the model using two set operations, union and subtraction.

This method facilitates the creation of a complex pattern of hundreds of geometric features in minutes. All the features are accessible from the CAD system and can be manipulated individually if required by the user. This allows the industrial designer or architect to explore more alternatives by avoiding the tedious and time-consuming manual generation of these geometric patterns.

As human beings, we are inclined to detect patterns in our world. Maybe reality itself is not made of such patterns, but for good reasons we have evolved to distinguish them among other physical objects. We are even better judges when it comes to understanding if patterns break, which was a survival mechanism for our ancestors. Patterns are all around us in nature, and as any engineer or artist will tell you they manifest in our creations, such as memory chips and computers, dresses and pillows, car exteriors and free-form architectural surfaces. These structures differentiate one object from another and separate objects from their surrounding space. They add interest and depth and can be used to add a sense of realism to an element or an entire object.

With the advent of additive manufacturing (AM), the creation of complex and unlikely designs has become more affordable. This process saves time in many applications because complete assemblies can be arranged from the start through part consolidation. It is feasible to generate a final object in just one step, eliminating stages of production. This dissertation explores new methods for pattern generation as a designers tool, in the area of mechanical engineering for applications using CAD and extended to AM examples.

The study of pattern genesis in the industrial design setting or mechanical engineering domain is actually poor or non-existent at the moment. The majority of advancement in the domains of pattern generation and texture synthesis comes from the computer graphics and image editing arenas, where the goal is to implement pattern and texture synthesis algorithms that strike the best balance between quality, speed and simplicity. We borrow their ideas and present here relevant definitions that can be incorporated into the mechanical engineering paradigm.

Problems in geometry can be solved using tools and techniques borrowed from linear algebra, integral calculus, differential calculus and multilinear algebra. The goal of discrete differential geometry is to connect classical mathematical structures embedded on smooth surfaces and curves, to polygons, meshes and simplicial complexes. Discrete differential geometry motivates efficient algorithms to tackle problems in computer graphics and architectural geometry.

Pixel-based methods typically synthesize a pattern or texture by finding and copying pixels with the most similar local neighborhood as the input texture into the synthesized pattern or texture. The method aims at preserving as much local structure as possible and produces good results for a wide variety of synthetic and real-world textures. These procedures initialize the output image with random noise to capture the random aspect found in natural structures and patterns. The techniques seed the algorithm with sufficient entropy and the rest of the synthesis process centers on transforming the output towards the input image.

## Comments

## Anthony

Do we have a example of this process in Grasshopper for the member’s?

## rezae

Hi Anthony,

There are plenty of different implementation in our examples, like the pattern category.

Here is a good example that I found in grasshopper3d

https://www.grasshopper3d.com/group/hoopsnake/forum/topics/lloyds-algorithm