Interpolation functions

Throughout the code NEML uses interpolation functions to represent model parameters that depend on temperature or, occasionally, some other variable. A NEML interpolation function is a scalar function of a single variable $f\left( x \right)$ .

The interface must also provide the first derivative of the function with respect that the variable.

These interpolation functions are used throughout NEML in place of scalar parameters. Except where noted, currently only for the InterpolatedIsotrpicHardeningRule, interpolation functions are used to give the parameter as a function of temperature. A constant parameter, e.g. one that does not depend on temperature can be expressed by using a ConstantInterpolate object.

Interpolate

The interface for all interpolate objects.

class Interpolate : public neml::NEMLObject 

Base class for interpolation functions.

Subclassed by neml::ConstantInterpolate, neml::ExpInterpolate, neml::GenericPiecewiseInterpolate, neml::MTSInterpolate, neml::MTSShearInterpolate, neml::PiecewiseLinearInterpolate, neml::PiecewiseLogLinearInterpolate, neml::PiecewiseSemiLogXLinearInterpolate, neml::PolynomialInterpolate, neml::PowerLawInterpolate

Public Functions

Interpolate(ParameterSet &params)

virtual double value(double x) const = 0: Returns the value of the function.

virtual double derivative(double x) const = 0: Returns the derivative of the function.

double operator()(double x) const: Nice wrapper for function call syntax.

bool valid() const: Is the interpolate valid?

ConstantInterpolate

Expresses a constant parameter

$f\left( x \right) = C.$

class ConstantInterpolate : public neml::Interpolate 

A constant value.

Public Functions

ConstantInterpolate(ParameterSet &params): The parameter is the constant value!

virtual double value(double x) const: Returns the value of the function.

virtual double derivative(double x) const: Returns the derivative of the function.

Public Static Functions

static std::string type(): Type for the object system.

static ParameterSet parameters(): Create parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Create object from a ParameterSet.

GenericPiecewiseInterpolate

A general piecewise interpolate. If $x \le p_i$ then it calls interpolation function $f_i$ . For $x > p_i$ it calls interpolation function $f_{i+1}$ .

class GenericPiecewiseInterpolate : public neml::Interpolate 

Generic piecewise interpolation.

Public Functions

GenericPiecewiseInterpolate(ParameterSet &params)

virtual double value(double x) const: Returns the value of the function.

virtual double derivative(double x) const: Returns the derivative of the function.

Public Static Functions

static std::string type(): Type for the object system.

static ParameterSet parameters(): Create parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Create object from a ParameterSet.

PolynomialInterpolate

Polynomial interpolation of the type

$f\left( x \right) = \sum_{i=1}^{n}c_{i}x^{i-1}.$

The polynomial coefficients are given starting with the highest order, like in the numpy library.

class PolynomialInterpolate : public neml::Interpolate 

Simple polynomial interpolation.

Public Functions

PolynomialInterpolate(ParameterSet &params): Input is the coefficients of the polynomial, from highest to lowest order.

virtual double value(double x) const: Returns the value of the function.

virtual double derivative(double x) const: Returns the derivative of the function.

Public Static Functions

static std::string type(): Type for the object system.

static ParameterSet parameters(): Create parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Create object from a ParameterSet.

PiecewiseLinearInterpolate

Piecewise linear interpolation where parameters give the x and y coordinates of the endpoints of each linear segment. For $x_{i} \le x \le x_{i+1}$

$f\left( x \right) = \frac{y_{i+1} - y_{i}}{x_{i+1} - x_{i}} \left(x - x_i \right) + y_i.$

For $x < x_1$ the function returns $y_1$ and for $x > x_n$ the function returns $y_n$ .

class PiecewiseLinearInterpolate : public neml::Interpolate 

Piecewise linear interpolation.

Public Functions

PiecewiseLinearInterpolate(ParameterSet &params): Parameters are a list of x coordinates and a corresponding list of y coordinates

virtual double value(double x) const: Returns the value of the function.

virtual double derivative(double x) const: Returns the derivative of the function.

Public Static Functions

static std::string type(): Type for the object system.

static ParameterSet parameters(): Create parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Create object from a ParameterSet.

PiecewiseLogLinearInterpolate

Exactly like PiecewiseLinearInterpolate except the interpolation in y is done in log space. The input y values are given as $y_i = \ln{v_i}$ where $v_i$ is the actual value of the function at $x_i$ . For $x_{i} \le x \le x_{i+1}$

$f\left( x \right) = \exp{\left(\frac{y_{i+1} - y_{i}}{x_{i+1} - x_{i}} \left(x - x_i \right) + y_i\right)}.$

For $x < x_1$ the function returns $\exp{\left(y_1\right)}$ and for $x > x_n$ the function returns $\exp{\left(y_n\right)}$ .

class PiecewiseLogLinearInterpolate : public neml::Interpolate 

Piecewise loglinear interpolation.

Public Functions

PiecewiseLogLinearInterpolate(ParameterSet &params): Similar to piecewise linear interpolation except the y coordinates are given as ln(y) and the interpolation is done in log space

virtual double value(double x) const: Returns the value of the function.

virtual double derivative(double x) const: Returns the derivative of the function.

Public Static Functions

static std::string type(): Type for the object system.

static ParameterSet parameters(): Create parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Create object from a ParameterSet.

MTSShearInterpolate

The shear modulus interpolation used in the Mechanical Threshold Stress [MTS1999] flow stress model

$f\left( x \right) = V_0 - \frac{D}{e^{T_0 / x} - 1}.$

class MTSShearInterpolate : public neml::Interpolate 

The MTS shear modulus function proposed in the original paper.

Public Functions

MTSShearInterpolate(ParameterSet &params): Interpolation using the MTS model form f(x) = V0 - D / (exp(T0 / x) - 1)

virtual double value(double x) const: Returns the value of the function.

virtual double derivative(double x) const: Returns the derivative of the function.

Public Static Functions

static std::string type(): Type for the object system.

static ParameterSet parameters(): Create parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Create object from a ParameterSet.

Helper Functions

std::vector<std::shared_ptr<Interpolate>> neml::make_vector(const std::vector<double> &iv): A helper to make a vector of constant interpolates from a vector.

std::vector<double> neml::eval_vector(const std::vector<std::shared_ptr<Interpolate>> &iv, double x): A helper to evaluate a vector of interpolates.

std::vector<double> neml::eval_deriv_vector(const std::vector<std::shared_ptr<Interpolate>> &iv, double x): A helper to evaluate the derivative of a vector of interpolates.