Using Extra Dimensions
Extra dimensions can be used to extend a standard geometry with additional spatial dimensions. Using extra dimensions it is possible, in principle, to solve PDEs in any number of independent variables, beyond 3D and time.
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Adding Extra Dimensions to a Model
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Definitions
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To add an extra dimension, right-click the Global node. See Adding Extra Dimensions to a Model. The settings for the Extra Dimension node are the same as for the Component node, except it has a unique Name.
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From the Definitions node, attach the extra dimensions to a selection in the base geometry. See Attached Dimensions.
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As required, add and define Points to Attach and Integration Over Extra Dimension nodes.
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Adjust the settings for these topics: Defining Equations and Variables on Extra Dimensions, Selections in the Geometry, and Plotting Results in Extra Dimensions.
Defining Equations and Variables on Extra Dimensions
To define equations in the product geometry formed by an Attached Dimensions feature, add a Weak Contribution (PDEs and Physics) feature to any physics. In the Selection section, select an extra dimension attachment feature in the Extra dimensions to attach table, and make a selection of geometric entities in the base geometry and in each attached extra dimension.
By default, there are no dependent variables defined in the extra dimensions. To define dependent variables, add an Auxiliary Dependent Variable subfeature, and then select the geometric entities in the base and extra dimension geometries where it should be defined.
Constraints in the extra dimensions can be defined by using Pointwise Constraint or Weak Constraint features with a selection in the product geometry.
Selections in the Geometry
Whenever an extra dimension geometry has been attached using an Attached Dimensions feature, an Extra dimensions to attach list displays in the selection section for features that support selection in the product geometry. By default the extra dimension attachment is set to None.
If the Extra dimensions to attach setting is changed to one of the Attached Dimension features, additional inputs appear for each attached extra dimension geometry. Use these to choose the geometric entity level and the geometric entities to select in each extra dimension.
Features that currently support selection in the product geometry are Variables, Weak Contribution (PDEs and Physics), Auxiliary Dependent Variable, Pointwise Constraint, and Weak Constraint.
Plotting Results in Extra Dimensions
A solution obtained by means of extra dimensions can be plotted in several ways:
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A “horizontal” section through the product geometry can be plotted by using one of the atxd operators. For example, if a 2D extra dimension has the tag xdim, the operator xdim1.atxd2(x0,y0,expr) evaluates expr at a point in the product geometry, defined by the coordinates (x0,y0) in the extra dimension geometry.
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Integrals over sections through the product geometry can be computed by using operators defined by Integration Over Extra Dimension features. For example, if an integration operator called xdintop1 has been defined, xdim1.xdintop1(expr) integrates expr over sections through the product geometry corresponding to the operator’s selection of geometric entities in the extra dimension geometry.
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It is also possible to make plots to plot along “vertical” sections through the product geometry. With Datasets, select the extra dimension as Component. Then, for example, if the base geometry is in 3D and the extra dimension’s name is xdim1, evaluate comp1.atxd3(x0,y0,z0,expr), where (x0, y0, z0) define the coordinates of a point in the base geometry.
Evaluation of Variables in Extra Dimensions
When working with extra dimensions, variables can be defined on the base geometry, on an extra dimension, or on a product geometry formed from the base geometry and one or several extra dimensions.
When evaluating a variable v in a product geometry, the rules for resolving the correct definition of v are as follows:
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If the variable is defined in the product geometry, this definition is used.
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Otherwise, if the variable has a definition in the base geometry, the variable is evaluated in the base geometry.
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Finally, if the variable is not defined in the product geometry or base geometry, but is defined in one of the extra dimensions, then it is evaluated in this extra dimension.
Naming of Partial Derivative in Extra Dimensions
Partial derivatives of dependent variables defined on a product of geometric entities of full dimension are formed by appending a coordinate name from the base geometry or one of the extra dimensions. For example, if u is a dependent variable, the coordinates in a 2D base geometry are called x and y; the coordinates in a 2D extra dimension are called x1 and y1; and the partial derivatives with respect to those coordinates are called ux, uy, ux1, and uy1, respectively. However, if the dependent variable is defined on an entity of lower dimension in either the base geometry or the extra dimension, insert the character T between the dependent variable name and the coordinate name. For example, if u is defined on the product of a domain in the base geometry and a boundary in the extra dimension (or vice versa), the partial derivatives are called uTx, uTy, uTx1, and uTy1.
Second derivatives follow the same pattern; for example, you can use uxx1, if u is defined on a product of entities of full dimension, or uTxx1, if u is defined on a product of entities of lower dimension.
You can also use the d and dtang operators to evaluate the partial derivatives of dependent variables defined in a product geometry.
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Example: Solving Poisson’s Equation in a Cylinder by Means of Extra Dimensions
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Results Analysis and Plots
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Working with Geometric Entities
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Creating Named Selections