4D fields

4D fields are formula fields or table fields whose independent domain includes both spatial coordinates and a single non-spatial variable like time, temperature, or frequency. Mathematically, 4D fields allow you to model functional relationships of the form:

f = f(x1,x2,x3,α)

where x1, x2, and x3 represent a set of Cartesian, cylindrical, spherical, or parametric spatial coordinates, and α represents a non-spatial variable like time, temperature, or frequency.

You can use 4D fields to define the following boundary conditions in solver environments other than Simcenter Samcef:

• Acceleration

• Heat flux

• Heat generation

• Nodal pressure

• Pressure

• Radiation

• Temperature

• Convection

• Enforced displacement

• Thermal constraints

In the Simcenter Samcef solver environment, you can use 4D fields to only define the following boundary conditions:

• Temperature load

• Pressure

• Degree of cure

• Glass transition temperature

4D formula fields

If the 4D functional relationship can be expressed as a closed-form mathematical relationship, you can create the 4D field as a formula field. When the software evaluates the formula field for a given set of spatial and non-spatial values, it simply evaluates the expression that relates the independent and dependent domains at the given values.

For example, suppose the independent and dependent domains are related by f = 4x1x3+2x22-2α. The software calculates the dependent domain value at (2,0,5,3) to be 4(2)(5)+2(02)-2(3) = 34.

Because the evaluation is direct, formula fields are not subject to interpolation errors.

You can define a 4D formula field where one or two of the spatial coordinates are not explicitly included in the expression that relates the independent and dependent domains. Mathematically, these functional relationships are expressed as:

f = f(x1,α)

and

f = f(x1,x2,α)

When the software evaluates such formula fields, the value calculated from the expression is used for all values of the missing variables.

For example, suppose the independent and dependent domains are related by f = 3x1x2+7α. The x3 spatial variable is missing from the expression. The software calculates the dependent domain value at (1,2,3,1) to be 3(1)(2)+7(1) = 13. Similarly, the software calculates the dependent domain value at (1,2,4,1) to also be 3(1)(2)+7(1) = 13.

4D table fields

If the 4D functional relationship can be approximated by a set of tabular data that relates the independent and dependent domains, you can create the 4D field as a table field. When the software evaluates the table field for a given set of spatial and non-spatial values at which a tabular data point does not exist, it interpolates the tabular data to obtain the corresponding dependent domain value.

You can select the interpolation method that the software uses to look up values. The interpolation methods applicable to 4D table fields are:

• Nearest neighbor and approximate nearest neighbor methods

• Renka’s modified Shepard method

• Inverse distance weighting method

These interpolation methods treat the non-spatial independent variable as a spatial independent variable in the interpolation calculations. This is true even though the dimensions of the non-spatial independent variable typically differ from the dimensions of the spatial independent variables.

For example, suppose the lookup point is (x1,x2,x3,α) = (3,3,0,0.5), a tabular data point exists at (4,4,0,1), and the inverse weighting method is selected as the interpolation method. The distance from the data point to the lookup point that the software calculates is [(4–3)2+(4–3)2+(0–0)2+(1–0.5)2]1/2 = 1.500, whereas the actual physical distance between these two points is [(4–3)2+(4–3)2+(0–0)2]1/2 = 1.414.

If the table lookup is only at locations where a data point is defined, the 4D interpolation effectively becomes an interpolation over only the non-spatial independent variable. For such a case, errors arising from treating the non-spatial independent variable as a spatial independent variable do not occur. However, for all other cases, as a best practice you should do one of the following:

• If you can relate the independent and dependent domains with a closed-form mathematical relationship, create a 4D formula field instead of a 4D table field.

• If you cannot relate the independent and dependent domains with a closed-form mathematical relationship, create a table of fields from the 4D table field.For more information, see Creating a table of fields from an existing field.

If a 4D formula field or a table of fields is not a viable alternative, and the table lookup is not exclusively at locations where a tabular data point is defined, you can minimize 4D interpolation errors as follows:

• Define enough tabular data points to ensure that a tabular data point is in close spatial and non-spatial proximity to any lookup point.

• Use interpolation sub-options like Approximate Nearest Neighbor Tolerance and Interpolate On to limit the distance from the lookup point at which a tabular data point can be located and still be included in the interpolation calculation.

How do I

Create a 4D field

Define a boundary condition using a 4D field

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4D fields, Simcenter 3D 2021.1 Series

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