## Applying Translations, Scaling, Reflection, and Rotation via the Deformed Geometry and Moving Mesh Interfaces

The *Deformed Geometry* and *Moving Mesh* interfaces are useful for modeling situations where a domain is deforming, as well as when it is moving in a completely prescribed way, either as a function of time or as a function of a model parameter in a parametric sweep or auxiliary sweep. Using these interfaces usually avoids any remeshing of the geometry or solving of additional equations.

### The Moving Mesh and Deformed Geometry Interfaces

In most cases, the *Moving Mesh* interface is the appropriate interface to use, since in this interface, the material, and solution, moves with the spatial deformation. The *Deformed Geometry* interface, on the other hand, implies that material is being added and removed from a domain as it translates through space, and is appropriate for models involving the *Solid Mechanics* interfaces, which themselves compute a deformation of the solid material as defined on the *Material* frame. It is not possible to solve for both *Solid Mechanics* and *Moving Mesh* within the same domain. Another way to keep track of this is in terms of these three frames:

*Geometry*frame coordinates, default , are fixed in the CAD geometry in 3D.*Material*frame coordinates, default , are fixed in the material in solid domains. A difference between*Material*and*Geometry*frame coordinates may be induced by*Deformed Geometry*or*Shape Optimization*features in the model. The difference represents a different configuration of the material, i.e., effectively a different shape of solid objects.*Spatial*frame coordinates, default , are fixed in space. A difference between the*Spatial*and*Material*frame coordinates may be induced by*Moving Mesh*features or structural mechanics interfaces, which control the*Spatial*frame. The difference represents a displacement of the material.

To verify which frame is being used for making plots, use the *Frame* setting within a *Dataset* feature, as shown in the screenshot below. Each plot group can also plot the edges of the dataset, in a selected frame, as shown below.

*Screenshot showing how to select the frame used to plot results.*

*Within each plot group, it is also possible to plot the dataset edges, using different frames.*

### Modeling Translations

To model the translation of a domain, such as for the movement of a part in space, define the expression of a translation curve in space: . Use these expressions within a *Moving Mesh* > *Prescribed Deformation* feature such that every point within the domain is offset from its original location by the same amount.

*Using the* Prescribed Deformation *feature to implement translation.*

For example, to move a domain along a helical spiral path centered along a line parallel to the *Z*-axis and offset from its original position, use these expressions for the prescribed deformation:

dx = `(1[m]+t*0.25[m/s])*cos(2*pi*t*1[Hz])+1[m]`

dy = `(1[m]+t*0.5[m/s])*sin(2*pi*t*1[Hz])+0.5[m]`

dz = `t*1[m/s]`

This translation example is shown in the above screenshot and implemented in the exercise file.

### Modeling Scaling & Reflection

To model a domain that is expanding or contracting about the global Cartesian coordinate system, it is necessary to define a scale factor matrix and a point to scale about. The scale factor matrix, , is a diagonal matrix with entries corresponding to scaling the Cartesian directions. For example:

will leave the *x*-dimensions unchanged, scale the *y*-dimension up by two, and reduce the *z*-dimension by half.

It is also necessary to define the point about which the domain is scaled, , and this defines the deformation field of the domain:

where is the identity matrix.

So, for example, to scale a part about the point 1,2,3 by the above scale factor matrix, prescribe a deformation of:

dx = `(1-1)*(X-1[m])`

dy = `(2-1)*(Y-2[m])`

dz = `(0.5-1)*(Z-3[m])`

It is also possible to make the terms functions of time or other model parameters. An example is implemented in the exercise file.

### Modeling Rotations, Scaling, and Translations

Modeling of generalized rotation can become complex. Two cases are addressed here: rotation about a Cartesian axis, as well as a more general case of 3D rotation.

For rotations about one of the global Cartesian axes, it is sufficient to define a single rotation matrix, and the coordinate of the center of rotation. For example, consider rotating a part through an angle theta about a line parallel to the *Z*-axis and passing through the point . The rotation matrix about any line parallel to the *Z*-axis is:

And, to rotate about the point , use the deformation:

This is equivalent to using the *Rotating Domain* feature. An example is included in the exercise file.

For the more general case of a part rotated and moved arbitrarily in space, two coordinate systems need to be conceptually defined, and , at points and , as shown in the image below. Consider the first, , as algined with the part. The objective is to rotate and move the part to align the part coordinate system with .

*A part with a coordinate system that will be rotated and moved to align with a second coordinate system. The first, second, and third axes of these systems are shown in red, green, and blue, respectively.*

Both coordinate systems are defined in terms of two orthonormal vectors, & , with the third vector defined by the cross-product: . These vectors define a transformation matrix:

This transformation matrix will rotate the part from one coordinate system to another. The above matrix is equivalent to applying a rotation that rotates the part from into the global Cartesian CS. This rotation should be done about , and then, to rotate from the global Cartesian to and move to point , use the following expression for deformation:

It is also possible to include a scaling. To do so in the coordinate system, use a combined scaling, rotation, and translation via an expression of:

An example of this is in the exercise file.

*Translation and anisotropic scaling applied, as well as rotation, in terms of two coordinate systems. The original undeformed part is shown in wireframe.*

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