expression

The expression command can be used to define mathematical expressions that can be later used to customise relevant material models.

The expression command essentially parses a block of plain text into mathematical expressions. The parsing and evaluation functionalities are powered by exprtk.

The exprtk generates an AST from the input text. The evaluation of the expression has an on par performance with the native C++ code.

But a customisable expression allows for more flexibility and it is easier to experiment different functions.

Syntax

Simple Scalar-valued Expression

expression SimpleScalar (1) (2) (3)
# (1) int, unique expression tag
# (2) string, input variable list
# (3) string, expression string or name of file containing expression

This evaluates a scalar-valued function $f(x_1,x_2,\cdots,x_n)$. The input arguments can be either scalars or vectors. For example, "x_1|2|x_2" represents two variables: 1) $x_1$ is a vector of size 2 and 2) $x_2$ is a scalar. The function value is returned as a result. The expression should be in the form of, for example, $x_1^2+x_2^2$.

This form supports automatic computation of derivatives. Thus, it can be used in models that require automatic computation of derivatives.

Simple Vector-valued Expression

expression SimpleVector (1) (2) (3) (4)
# (1) int, unique expression tag
# (2) string, input variable list
# (3) string, output variable list
# (4) string, expression string or name of file containing expression

This evaluates a vector-valued function $y:=f(x_1,x_2,\cdots,x_n)$. An assignment must be used to explicitly assign values to the output variables. The input arguments can be either scalars or vectors. The output argument should be a vector, the size shall be given such that $y|3$ represents a vector of size 3.

This form does not support automatic computation of derivatives. Thus, it cannot be used in models that require automatic computation of derivatives.

Remarks

1. The variables in the expression need to be explicitly given as the arguments. It can be either unquoted or quoted by ".

2. Multiple variables must be separated by vertical bar |.

3. The expression can be given as a string or as a file name. The program tries to load the file first. If the file does not exist, the program tries to parse the string as an expression.

4. No whitespace (including ,, , etc.) is allowed in any arguments.

exprtk provides very powerful functionalities. It is possible to apply logics and control flows in the expression.

Example

Say, for example, one is going to define a uniaxial nonlinear elastic model that uses the following stress-strain relationship:

$$\sigma=\varepsilon^3+\varepsilon$$

Then the expression can be defined as

expression SimpleScalar 1 x x^3+x

Here, we use x as the free variable and the corresponding expression is x^3+x.

For the unixial nonlienar elastic model, the input is strain, which maps to x, the output maps to stress.

Alternatively, the expression can be stored in a file and loaded by the program.

x^3+x

Then the expression can be defined as

expression SimpleScalar 1 x cubic.txt

For complex expressions, it is convenient to use the file format.

To use the expression, here, CustomElastic1D is used.

# using expression 1
material CustomElastic1D 1 1

materialTest1D 1 1E-2 200

exit

The response can be tested by material tester.

Result

cubic response