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Local Basis

Each finite element includes a kind of local basis in terms of ansatz functions. These ansatz functions need to be evaluated in the parameter domain of the finite element.

Interface

In the exported module dune-localfefunctions, we provide a thin wrapper with basic caching functionality for dune bases. For the definitions of bases, refer Chapter 8.2.1 of the Dune book1. Thus Dune::CachedLocalBasis (#include <dune/cachedlocalBasis/cachedlocalBasis.hh>) provides the following interface: 

Dune::LocalBasis(const DuneLocalBasis& p_basis);// (1)!
void evaluateFunction(const DomainType& local, Eigen::VectorX<RangeFieldType>& N);
void evaluateJacobian(const DomainType& local,Eigen::Matrix<RangeFieldType, Eigen::Dynamic, gridDim>& dN);
void evaluateFunctionAndJacobian(const DomainType& local,Eigen::VectorX<RangeFieldType>& N,
                                 Eigen::Matrix<RangeFieldType, Eigen::Dynamic, gridDim>& dN);

void evaluateSecondDerivatives(const DomainType& local, Eigen::Matrix<RangeFieldType, Eigen::Dynamic, gridDim*(gridDim + 1) / 2>& dN);                                 

const Eigen::VectorX<RangeFieldType>& evaluateFunction(const unsigned int& integrationPointIndex);
const Eigen::Matrix<RangeFieldType, Eigen::Dynamic, gridDim>& evaluateJacobian(const unsigned int& integrationPointIndex);
const Eigen::Matrix<RangeFieldType, Eigen::Dynamic, gridDim*(gridDim + 1) / 2>& evaluateSecondDerivatives(const unsigned int& integrationPointIndex);
auto viewOverIntegrationPoints();// (2)!
auto viewOverFunctionAndJacobian();// (3)!

template <typename IntegrationRule, typename... Ints>
void bind(IntegrationRule&& p_rule, Derivatives<Ints...>&& ints);

bool isBound(int i) const;// (4)!
const Dune::QuadraturePoint<DomainFieldType, gridDim>& indexToIntegrationPoint(int i) const;// (5)!

  1. The constructor only accepts a local basis that satisfies the concept Concepts::DuneLocalBasis. In keeping with the spirit of duck-typing, this also allows for the use of a local basis from the Dune module.
  2. This returns a vector of structs containing the integration point and its index. Therefore, the syntax is usually for (const auto& [gpIndex, gp] : localFunction.viewOverIntegrationPoints()) {...}
  3. Returns a view over the ansatz functions and the ansatz function Jacobians at the integration points
  4. Checks if the i-th derivatives are bounded.
  5. Convert an integration point index to a full integration point.

The first two function calls, evaluateFunction and evaluateJacobian, can be used to calculate the function values \( N(\boldsymbol{\xi}) \) and the spatial derivatives \( N_{,\boldsymbol{\xi}}(\boldsymbol{\xi}) \). One must allocate the objects and pass them as mutable references. The same holds for evaluateSecondDerivatives.

In contrast to this, there are three other methods that receive an integration point index. These methods return a const reference to the evaluated ansatz function values and their first and second derivatives.

This functionality is dependent on a previous call to bind(...). This binds the local basis to one quadrature rule and caches the passed bindDerivatives(..). An error is thrown if evaluateFunction(const unsigned int& integrationPointIndex) is called before binding. Finally, to bind to an integration rule and cache the value and ansatz function Jacobian, one would use the following syntax:

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using LocalFEType = LagrangeSimplexLocalFiniteElement<double,double,dim,2>;
LocalFEType localFE;

Dune::CachedLocalBasis(localFE.localBasis());
const auto& rule = Dune::QuadratureRules<double, Traits::mydim>::rule(localView.element().type(), order);
localBasis.bind(rule, bindDerivatives(0, 1));

for (const auto& [gpIndex, gp] : localBasis.viewOverIntegrationPoints()) {
  const auto& N = localBasis.evaluateFunction(gpIndex);
  const auto& dN = localBasis.evaluateJacobian(gpIndex);
}

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using LocalFEType = LagrangeSimplexLocalFiniteElement<double,double,dim,2>;
LocalFEType localFE;

Dune::CachedLocalBasis(localFE.localBasis());
const auto& rule = Dune::QuadratureRules<double, Traits::mydim>::rule(localView.element().type(), order);
Eigen::VectorXd N;
Eigen::Matrix<double, Eigen::Dynamic, gridDim> dN;

for(auto& gp : rule){
  localFunction.evaluateFunction(gp.position(), N); 
  localFunction.evaluateJacobian(gp.position(), dN);
  localFunction.evaluateFunctionAndJacobian(gp.position(), N, dN);// (1)! 
}
  1. Alternative to the two lines above (Line 6 and 7)

  1. Oliver Sander. DUNE—The Distributed and Unified Numerics Environment. Volume 140. Springer Nature, 2020. doi:10.1007/978-3-030-59702-3. ↩