Meshing and discretization

Texture mapping has long existed as a means of augmenting 3D surface data with additional detail (Heckbert 1986; Blinn 1978). This additional detail takes the form of arbitrary multi‐dimensional parameters: surface color, specular reflection, normal vector perturbation, and so on. The mapping of these parameters onto arbitrary 3D surfaces demonstrates a useful characteristic for multi‐scalar modelling: that of decoupled models and datasets. Texture maps can be of arbitrary resolution, irrespective of the target 3D geometry that they are being mapped onto. Similarly, this makes them interchangeable.

Although the use of texture maps for encoding additional parameters on modelled timber surfaces is useful, discretization of the 3D model into elements allows a similar encoding of parameters which embeds their distribution into the topology of the model. Discretization also is not constrained by the grid‐like topology of image maps ‐ a common way of mapping texture data. Further, this division of the 3D model into discrete elements allows an almost one‐to‐one interfacing with material and structural simulation techniques, such as thefinite element method(FEM) (Larson and Bengzon 2013).

In a more general sense, discretization can be used as a method for moving in‐between different scales of modelling in a multi‐scalar approach.

Discretizing a model at one scale represents a refinement of information at a lower scale, such as described by Nicholas, Zwierzycki, and Ramsgaard Thomsen (2015), along with increasing the amount of data being considered and processed. Thus meshing is only one form of discretization that is conveniently applicable to the encoding of heterogeneous properties in 3D surface models. Beyond the individual timber plank, the glulam blank can be discretized into its constituent lamellae, and the glulam structure can be discretized into its constituent glulam components. The discretization of the glulam blank is presented later on as an extension of this approach.

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Fig. 4.2:Discretizing a timber component. A timber surface with highly varying properties (top). A discretization technique decomposes the rectangular boundary into triangle elements based on a variance map (middle). Element density corresponds to areas of higher material variability (bottom).

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Fig. 4.3:A proportional subdivision (top), grid‐like subdivision (middle), and an adaptive triangle subdivision (bottom) of a 2‐dimensional timber plank.

Since timber is made up of individual wood cells and fibres, a complete discretization of a timber plank model would result in an element for each cell and fibre: an increase in resolution that matches the material resolution.

In typical use cases, however, local properties can only be inferred at lower resolutions, meaning microscopic properties are averaged into more abstracted indications of local material property.

Types of meshing

This probe took the timber plank as its starting point due to its simplicity:

projected to two dimensions, the plank is a simple rectangle and can be represented as a mesh with a single face. As planks are linear elements, the proportion of the rectangle is several times wider as it is high (Fig. 4.3).

Proceeding from this, the most basic discretization consists of dividing the rectangle into smaller faces with roughly equal proportions. This first and coarsest technique creates a differentiation of properties along the length of the plank.

Next, the resolution of this discretization in increased by further subdividing the faces in both dimensions: a grid‐like discretization. Again, this is well‐suited to a rectangular plank because it is orthogonal to begin with and is topologically trivial. A problem arises once the plank is no longer purely rectangular ‐ such as when it is cut at an angle or has cut‐outs. Further, a grid‐like discretization yields elements that are all similarly sized and evenly spaced. This can misrepresent the underlying timber surface which can have large areas of even properties (clearwood) interspersed with small areas of large property deviations (knots). These small interruptions

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nevertheless have a large impact on the performance and behaviour of the plank. However, subdividing the whole plank to a much higher resolution to capture these small areas also means that the large and even areas have a mass of redundant data.

A third method of discretization is therefore triangle‐based and adaptive which allows an uneven distribution of model elements across a timber surface. Higher‐resolution elements can be applied to areas of the timber plank that are varying the most and, conversely, lower‐resolution elements can be applied to larger areas of more even properties. This comes at the cost of the uniform and easily traversable topology generated by the preceding two methods, since the triangle elements are not organized in a grid.

To implement this third method, the probe uses a modified implementation of the Poisson disk sampling method by Bridson (2007) and a Delaunay triangulation of the resulting points (Fig. 4.2). The sampling disk radius is driven by a processed image map of the timber surface, which makes it possible to relate the sample density with the specific timber surface. For this probe, the intensities of the image values are used, blurred to remove high‐frequency information below the minimum sampling distance, and normalized. The resultant values drive the disk radii between a specified minimum and maximum distance, arbitrarily chosen to illustrate the effect.

Although the image gradient would be a better candidate to use for driving the disk radii, this method still correctly allocates finer elements around areas of high variation such as knots, and thus demonstrates the adaptivity of this method.

Glue‐laminated components

Extending this meshing approach to glue‐laminated components requires the definition of different regions of mesh elements that correspond to the constituent lamellae or timber planks that are glued together. A similar method is used as before, however with the addition of a property per resultant mesh element that identifies which part of the glue‐laminated component it belongs to. A laminated component can be therefore construed as distinct groupings of mesh elements that represent the separate timber elements within the laminated assembly.

This probe uses Otto Hetzer’s patent no. 163144 as a test case (Fig. 4.5) since it presents the simple case of a bent plank laminated in‐between two halves of an un‐bent block of timber. An exploratory physical prototype illustrates the principle (Fig. 4.4).

To implement this case study, the .NET implementation of the Triangle library

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Fig. 4.4:A physical probe similar to Otto Hetzer’s patent no. 163144.

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Fig. 4.5:Adaptive meshing of Otto Hetzer’s patent no. 163144. Each zone or group of elements represents a different constituent element. The boundaries between zones represent the glue boundaries.

by Shewchuk (1996) was adapted for use with the RhinoCommon API. The Triangle library creates constrained Delaunay triangulations from arbitrary point or segment input, while respecting boundaries of input polygons or regions. This allows different regions to be defined as polygons with shared boundaries ‐ corresponding to the glue‐line interfaces between laminated components ‐ which, after meshing, can be recovered as groupings of mesh elements. The constrained Delaunay method has the added advantage of responding to a variable distribution of input points and segments. This means that the thinner centre piece in Fig. 4.5 is discretized to a finer degree than the two surrounding pieces.

Fibre interfaces

Of particular interest in this move from individual timber plank to a

glue‐laminated component are the boundaries between the regions of mesh elements (interior) and between the mesh elements and the outside of the model (exterior) (Fig. 4.6). The interior boundaries between regions of mesh elements represent glue‐line boundaries, where the fibre directions of two separate timber pieces meet. Comparing the material direction of mesh elements on either side of this boundary yields important information about the quality of the glue‐line, since this is dependent on the orientation of fibres: effectiveness of adhesion is higher when the wood grains are parallel.

The exterior boundary ‐ between the mesh elements and the outside of the model ‐ yield useful information about the end‐grain exposure of the model, which, in turn, has important repercussions for the durability and structural

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v_n

v_n v_g

v_g

v_g1

v_g2

Fig. 4.6:Fibre boundaries in the discretized wood model. Thegreenarea shows where the surface normalv⃗nis perpendicular to the wood grain vectorv⃗g, meaning the surface is parallel to the grain direction ‐ an ideal scenario from a structural and durability perspective. Thebluearea shows the opposite case, where the surface normalv⃗nand grain vectorv⃗gare parallel, meaning the surface is cutting across the wood fibres. Theredarea shows a glue‐line boundary (red dashed line) and the two wood grain vectors (v⃗g1andv⃗g2) on either side of it. Comparing these two can give an indication of the effective adhesion between the two laminated regions.

performance of the timber component. Comparing the normal vector of this exterior boundary with the material direction in the neighbouring mesh element gives the amount of end‐grain present: if the normal is perpendicular to the material direction, then the fibres are parallel to the boundary and there is no end‐grain; if the normal is parallel to the material direction, then the fibres are orthogonal to the boundary and there is full end‐grain exposure.

Representing end‐grain and fibre direction on the model boundary can therefore give an insight at a glance about issues of durability and structural performance, without even considering the fully discretized interior of the model.

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