funcy - automatic differentiation for biomechanical modeling

Installation

funcy is header-only and requires no installation. To make it available in the local cmake-registry you still may find it useful install funcy using cmake.

Install funcy via cmake

mkdir build
cd build
cmake .. -DFuncy_BuildTest=ON -DCMAKE_INSTALL_PREFIX=<install-prefix>
cmake --build . --target install

To include funcy in your project add this to your CMakeLists.txt:

find_package(Funcy REQUIRED)
target_link_libraries(<target> PRIVATE Funcy::Funcy)

Using funcy

The function with funcy:

#include <funcy/funcy.h>

int main()
{
    using namespace funcy;
    // qualified function to sqrt to distinguish from the c-function sqrt in math.h, commonly included in the c++-header cmath.
    const auto y = funcy::sqrt(1.);
    auto f = finalize( pow<3>(y) + sin(y) );

    std::cout << "f(3) = " << f(3) << std::endl;
    f.update(4);
    std::cout << "f'(4) = " << f.d1() << std::endl;
    std::cout << "f'''(4) = " << f.d3() << std::endl;
}

The function finalize simplifies usage of f, taking advantage of the fact that it depends on only one variable and is defined on a one-dimensional space. The following examples illustrate usage without finalize and with vector and matrix-valued arguments.

A model for nonlinear heat transfer :

#include <funcy/funcy.h>

template <class Vector>
auto heatModel(double c, double d, double u0, const Vector& du0)
{
    using funcy::variable;
    const auto u = variable<0>(u0);
    const auto du = variable<0>(du0);
    return (c+d*squared(u))*du;
}

int main()
{
    auto f = heatModel(c, d, u, du);
    // update function value
    f.template update<0>(std::make_tuple(u, du));

    std::cout << "f(u, du) = " << f() << std::endl;
    // derivative with respect to variable with id 0
    std::cout << "f'(4) = " << f.template d1<0>(std::make_tuple(v,dv)) << std::endl;
}

A complex biomechanical model for adipose tissue, based on isotropic and anisotropic matrix invariants :

#include <funcy/funcy.h>

template <class Matrix>
auto adiposeTissue(double cCells, double k1, double k2, double kappa,
                   const Matrix& M const Matrix& F, int n = dim<Matrix>())
{
    using funcy::finalize;
    using namespace funcy::linalg;

    auto aniso = kappa*i1(F) + (1-3*kappa) * i4(F,M) - 1;
    auto materialLaw = cCells*( i1(F) - n ) + (k1/k2) * ( exp(k2*squared(aniso)) - 1 );

    return finalize( materialLaw( strainTensor(F) ) );
}

int main()
{
    // update function value
    f.update(du);

    std::cout << "f(du) = " << f() << std::endl;
    std::cout << "f''(du)(dv,dw) = " << f.d2(dv,dw) << std::endl;
}

Citing

L. Lubkoll: FunG - Automatic differentiation for invariant-based modeling. Archive of Numerical Software, vol. 5, no. 1, 2017, pp. 169-192, DOI: 10.11588/ans.2017.1.27477

Documentation

Available here:

Compatibility

funcy can work with any type that satisfies basic arithmetic requirements as described in requirements. Particular support has been implemented for scalars, vectors and matrices, enabling funcy to be used with all popular matrix libraries.

Optimization Strategies

For details, and performance comparisons with other AD-libraries, see FunG - Automatic differentiation for invariant-based modeling.

Publications:

Published:

• L. Lubkoll: FunG - Automatic differentiation for invariant-based modeling. Archive of Numerical Software, vol. 5, no. 1, 2017, pp. 169-192, DOI: 10.11588/ans.2017.1.27477
• L. Lubkoll, A. Schiela, M. Weiser: An affine covariant composite step method for optimization with PDEs as equality constraints.
  Opt. Meth. Softw., 32(5), pp. 1132-1161, 2016, (Preprint)
• L. Lubkoll: An Optimal Control Approach to Implant Shape Design: Modeling, Analysis and Numerics.
  Dissertation, Universität Bayreuth, 2015

Preprints:

• A. Schiela, M. Stöcklein: Optimal Control of Static Contact in Finite Strain Elasticity. 2018
• M. Schaller, A. Schiela, M. Stöcklein: A Composite Step Method with Inexact Step Comptations for PDE Constrained Optimization. 2018