SurfaceSlidingConstrains Struct Reference

Shape preserving constrains, i.e. nodes sliding on body surface. More...

#include <users_modules/basic_finite_elements/src/SurfaceSlidingConstrains.hpp>

Inheritance diagram for SurfaceSlidingConstrains:

Collaboration diagram for SurfaceSlidingConstrains:

Classes
struct	DriverElementOrientation
	Class implemented by user to detect face orientation. More...
struct	MyTriangleFE
struct	OpJacobian

Public Member Functions
MyTriangleFE &	getLoopFeRhs ()
MyTriangleFE &	getLoopFeLhs ()
MoFEMErrorCode	getOptions ()
	SurfaceSlidingConstrains (MoFEM::Interface &m_field, DriverElementOrientation &orientation)
virtual	~SurfaceSlidingConstrains ()=default
MoFEMErrorCode	setOperators (int tag, const std::string lagrange_multipliers_field_name, const std::string material_field_name, const double *alpha=nullptr)
MoFEMErrorCode	setOperatorsConstrainOnly (int tag, const std::string lagrange_multipliers_field_name, const std::string material_field_name)
Public Member Functions inherited from GenericSliding
	GenericSliding ()=default
virtual	~GenericSliding ()=default

Public Attributes
MoFEM::Interface &	mField
boost::shared_ptr< MyTriangleFE >	feRhsPtr
boost::shared_ptr< MyTriangleFE >	feLhsPtr
MyTriangleFE &	feRhs
MyTriangleFE &	feLhs
DriverElementOrientation &	crackFrontOrientation
double	aLpha

Detailed Description

Shape preserving constrains, i.e. nodes sliding on body surface.

Derivation and implementation of constrains preserving body surface, i.e. body shape and volume.

The idea starts form observation that body shape can be globally characterized by constant calculated as volume over its area

\[ \frac{V}{A} = C \]

Above equation expressed in integral form is

\[ \int_\Omega \textrm{d}V = C \int_\Gamma \textrm{d}S \]

where notting that,

\[ \frac{1}{3} \int_\Omega \textrm{div}[\mathbf{X}] \textrm{d}\Omega = C \int_\Gamma \textrm{d}S \]

and applying Gauss theorem we get

\[ \int_\Gamma \mathbf{X}\cdot \frac{\mathbf{N}}{\|\mathbf{N}\|} \textrm{d}\Gamma = 3C \int_\Gamma \textrm{d}S. \]

Drooping integrals on both sides, and linearizing equation, we get

\[ \frac{\mathbf{N}}{\|\mathbf{N}\|} \cdot \delta \mathbf{X} = 3C - \frac{\mathbf{N}}{\|\mathbf{N}\|}\cdot \mathbf{X} \]

where \(\delta \mathbf{X}\) is displacement sub-inctrement. Above equation is a constrain if satisfied in body shape and volume is conserved. Final form of constrain equation is

\[ \mathcal{r} = \frac{\mathbf{N}}{\|\mathbf{N}\|}\cdot \mathbf{X} - \frac{\mathbf{N_0}}{\|\mathbf{N_0}\|}\cdot \mathbf{X}_0 = \frac{\mathbf{N}}{\|\mathbf{N}\|}\cdot (\mathbf{X}-\mathbf{X}_0) \]

In the spirit of finite element method the constrain equation is multiplied by shape functions and enforce using Lagrange multiplier method

\[ \int_\Gamma \mathbf{N}^\mathsf{T}_\lambda \left( \frac{\mathbf{N}}{\|\mathbf{N}\|}\mathbf{N}_\mathbf{X}\cdot (\overline{\mathbf{X}}-\overline{\mathbf{X}}_0) \right) \|\mathbf{N}\| \textrm{d}\Gamma = \mathbf{0}. \]

Above equation is nonlinear, applying to it Taylor expansion, we can get form which can be used with Newton interactive method

\[ \begin{split} &\int_\Gamma \mathbf{N}^\mathsf{T}_\lambda \left\{ \mathbf{N}\mathbf{N}_\mathbf{X} + \left(\mathbf{X}-\mathbf{X}_0\right) \cdot \left( \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\xi}\right]\cdot\mathbf{B}_\eta - \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\eta}\right]\cdot\mathbf{B}_\xi \right) \right\} \textrm{d}\Gamma \cdot \delta \overline{\mathbf{X}}\\ = &\int_\Gamma \mathbf{N}^\mathsf{T}_\lambda \mathbf{N}\cdot(\mathbf{X}-\mathbf{X}_0) \textrm{d}\Gamma \end{split}. \]

Equation expressing forces on shape as result of constrains, as result Lagrange multiplier method have following form

\[ \begin{split} &\int_\Gamma \mathbf{N}^\mathsf{T}_\mathbf{X} \cdot \mathbf{N} \mathbf{N}_\lambda \textrm{d}\Gamma \cdot \delta\overline{\lambda}\\ + &\int_\Gamma \lambda \mathbf{N}^\mathsf{T}_\mathbf{X} \left( \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\xi}\right]\cdot\mathbf{B}_\eta - \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\eta}\right]\cdot\mathbf{B}_\xi \right) \textrm{d}\Gamma \delta\overline{\mathbf{X}}\\ = &\int_\Gamma \lambda \mathbf{N}^\mathsf{T}_\mathbf{X} \cdot \mathbf{N} \textrm{d}\Gamma \end{split} \]

Above equations are assembled into global system of equations as following

\[ \left[ \begin{array}{cc} \mathbf{K} + \mathbf{B} & \mathbf{C}^\mathsf{T} \\ \mathbf{C} + \mathbf{A} & 0 \end{array} \right] \left\{ \begin{array}{c} \delta \overline{\mathbf{X}} \\ \delta \overline{\lambda} \end{array} \right\}= \left[ \begin{array}{c} \mathbf{f} - \mathbf{C}^\mathsf{T}\overline{\lambda} \\ \overline{\mathbf{r}} \end{array} \right] \]

where

\[ \mathbf{C}= \int_\Gamma \mathbf{N}_\lambda^\mathsf{T} \mathbf{N} \cdot \mathbf{N}_\mathbf{X} \textrm{d}\Gamma, \]

\[ \mathbf{B}= \int_\Gamma \lambda (\mathbf{X}-\mathbf{X}_0)\cdot \left( \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\xi}\right]\cdot\mathbf{B}_\eta - \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\eta}\right]\cdot\mathbf{B}_\xi \right) \textrm{d}\Gamma \]

and

\[ \mathbf{A}= \int_\Gamma \mathbf{N}^\mathsf{T}_\lambda \left(\mathbf{X}-\mathbf{X}_0\right) \cdot \left( \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\xi}\right]\cdot\mathbf{B}_\eta - \textrm{Spin}\left[\frac{\partial\mathbf{X}}{\partial\eta}\right]\cdot\mathbf{B}_\xi \right). \]

Examples

mesh_smoothing.cpp.

Definition at line 323 of file SurfaceSlidingConstrains.hpp.

Constructor & Destructor Documentation

SurfaceSlidingConstrains()

SurfaceSlidingConstrains::SurfaceSlidingConstrains ( MoFEM::Interface &  m_field, DriverElementOrientation &  orientation  )

inline

Definition at line 405 of file SurfaceSlidingConstrains.hpp.

      : mField(m_field), feRhsPtr(new MyTriangleFE(m_field)),
        feLhsPtr(new MyTriangleFE(m_field)), feRhs(*feRhsPtr), feLhs(*feLhsPtr),
        crackFrontOrientation(orientation), aLpha(1) {
 
    ierr = getOptions();
    CHKERRABORT(PETSC_COMM_WORLD, ierr);
  }

~SurfaceSlidingConstrains()

virtual SurfaceSlidingConstrains::~SurfaceSlidingConstrains ( )

virtualdefault

Member Function Documentation

getLoopFeLhs()

MyTriangleFE& SurfaceSlidingConstrains::getLoopFeLhs ( )

inline

Definition at line 387 of file SurfaceSlidingConstrains.hpp.

{ return feLhs; }

getLoopFeRhs()

MyTriangleFE& SurfaceSlidingConstrains::getLoopFeRhs ( )

inline

Definition at line 385 of file SurfaceSlidingConstrains.hpp.

{ return feRhs; }

getOptions()

MoFEMErrorCode SurfaceSlidingConstrains::getOptions ( )

inline

Definition at line 392 of file SurfaceSlidingConstrains.hpp.

                              {
    MoFEMFunctionBegin;
    ierr = PetscOptionsBegin(PETSC_COMM_WORLD, "",
                             "Get surface sliding constrains element scaling",
                             "none");
    CHKERRQ(ierr);
    CHKERR PetscOptionsScalar("-surface_sliding_alpha", "scaling parameter", "",
                              aLpha, &aLpha, PETSC_NULL);
    ierr = PetscOptionsEnd();
    CHKERRQ(ierr);
    MoFEMFunctionReturn(0);
  }

setOperators()

MoFEMErrorCode SurfaceSlidingConstrains::setOperators ( int  tag, const std::string  lagrange_multipliers_field_name, const std::string  material_field_name, const double *  alpha = nullptr  )

inline

Definition at line 563 of file SurfaceSlidingConstrains.hpp.

                                                             {
    MoFEMFunctionBegin;
 
    if (alpha != nullptr) {
      aLpha = *alpha;
    }
 
    boost::shared_ptr<VectorDouble> active_variables_ptr(
        new VectorDouble(3 + 9));
    boost::shared_ptr<VectorDouble> results_ptr(new VectorDouble(3 + 9));
    boost::shared_ptr<MatrixDouble> jacobian_ptr(
        new MatrixDouble(3 + 9, 3 + 9));
 
    feRhs.getOpPtrVector().clear();
    feRhs.getOpPtrVector().push_back(new OpGetActiveDofsLambda(
        lagrange_multipliers_field_name, active_variables_ptr));
    feRhs.getOpPtrVector().push_back(new OpGetActiveDofsPositions<3>(
        material_field_name, active_variables_ptr));
    feRhs.getOpPtrVector().push_back(new OpJacobian(
        tag, lagrange_multipliers_field_name, active_variables_ptr, results_ptr,
        jacobian_ptr, crackFrontOrientation, false, aLpha));
    feRhs.getOpPtrVector().push_back(
        new OpAssembleRhs<3, 9>(lagrange_multipliers_field_name, results_ptr));
    feRhs.getOpPtrVector().push_back(
        new OpAssembleRhs<3, 9>(material_field_name, results_ptr));
 
    // Adding operators to calculate the left hand side
    feLhs.getOpPtrVector().clear();
    feLhs.getOpPtrVector().push_back(new OpGetActiveDofsLambda(
        lagrange_multipliers_field_name, active_variables_ptr));
    feLhs.getOpPtrVector().push_back(new OpGetActiveDofsPositions<3>(
        material_field_name, active_variables_ptr));
    feLhs.getOpPtrVector().push_back(new OpJacobian(
        tag, lagrange_multipliers_field_name, active_variables_ptr, results_ptr,
        jacobian_ptr, crackFrontOrientation, true, aLpha));
    feLhs.getOpPtrVector().push_back(new OpAssembleLhs<3, 9>(
        lagrange_multipliers_field_name, material_field_name, jacobian_ptr));
    feLhs.getOpPtrVector().push_back(new OpAssembleLhs<3, 9>(
        material_field_name, lagrange_multipliers_field_name, jacobian_ptr));
    feLhs.getOpPtrVector().push_back(new OpAssembleLhs<3, 9>(
        material_field_name, material_field_name, jacobian_ptr));
 
    MoFEMFunctionReturn(0);
  }

setOperatorsConstrainOnly()

MoFEMErrorCode SurfaceSlidingConstrains::setOperatorsConstrainOnly ( int  tag, const std::string  lagrange_multipliers_field_name, const std::string  material_field_name  )

inline

Definition at line 612 of file SurfaceSlidingConstrains.hpp.

                                                                 {
    MoFEMFunctionBegin;
 
    boost::shared_ptr<VectorDouble> active_variables_ptr(
        new VectorDouble(3 + 9));
    boost::shared_ptr<VectorDouble> results_ptr(new VectorDouble(3 + 9));
    boost::shared_ptr<MatrixDouble> jacobian_ptr(
        new MatrixDouble(3 + 9, 3 + 9));
 
    // Adding operators to calculate the left hand side
    feLhs.getOpPtrVector().clear();
    feLhs.getOpPtrVector().push_back(new OpGetActiveDofsLambda(
        lagrange_multipliers_field_name, active_variables_ptr));
    feLhs.getOpPtrVector().push_back(new OpGetActiveDofsPositions<3>(
        material_field_name, active_variables_ptr));
    feLhs.getOpPtrVector().push_back(new OpJacobian(
        tag, lagrange_multipliers_field_name, active_variables_ptr, results_ptr,
        jacobian_ptr, crackFrontOrientation, true, aLpha));
    feLhs.getOpPtrVector().push_back(new OpAssembleLhs<3, 9>(
        lagrange_multipliers_field_name, material_field_name, jacobian_ptr));
 
    MoFEMFunctionReturn(0);
  }

Member Data Documentation

aLpha

double SurfaceSlidingConstrains::aLpha

Definition at line 391 of file SurfaceSlidingConstrains.hpp.

crackFrontOrientation

DriverElementOrientation& SurfaceSlidingConstrains::crackFrontOrientation

Definition at line 389 of file SurfaceSlidingConstrains.hpp.

feLhs

MyTriangleFE& SurfaceSlidingConstrains::feLhs

Definition at line 386 of file SurfaceSlidingConstrains.hpp.

feLhsPtr

boost::shared_ptr<MyTriangleFE> SurfaceSlidingConstrains::feLhsPtr

Definition at line 382 of file SurfaceSlidingConstrains.hpp.

feRhs

MyTriangleFE& SurfaceSlidingConstrains::feRhs

Definition at line 384 of file SurfaceSlidingConstrains.hpp.

feRhsPtr

boost::shared_ptr<MyTriangleFE> SurfaceSlidingConstrains::feRhsPtr

Definition at line 382 of file SurfaceSlidingConstrains.hpp.

mField

MoFEM::Interface& SurfaceSlidingConstrains::mField

Definition at line 353 of file SurfaceSlidingConstrains.hpp.

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The documentation for this struct was generated from the following file:

• users_modules/basic_finite_elements/src/SurfaceSlidingConstrains.hpp