artisynth.core.materials

Class TensorUtils

java.lang.Object

artisynth.core.materials.TensorUtils

public class TensorUtils
extends java.lang.Object

Constructor Summary

Constructors

Constructor and Description
TensorUtils()

Method Summary

Modifier and Type	Method and Description
static void	addScaled4thPowerProduct(Matrix6d D, double s, Vector3d a) Adds the scaled tensor product
static void	addScaledIdentity(Matrix6d D, double s) Adds a scaled identity to a 4th order constitutive tensor represented as a 6 x 6 matrix.
static void	addScaledIdentityProduct(Matrix6d D, double s) Adds the scaled tensor product
static void	addSymmetricIdentityProduct(Matrix6d D, Matrix3dBase S) Adds the scaled tensor product
static void	addSymmetricTensorProduct(Matrix6d D, double s, Matrix3dBase A, Matrix3dBase B) Adds a scaled symmetric tensor product
static void	addSymmetricTensorProduct(Matrix6d D, Matrix3dBase A, Matrix3dBase B) Adds the symmetric tensor product
static void	addSymmetricTensorProduct4(Matrix6d D, double s, Matrix3dBase A, Matrix3dBase B) Adds the scaled symmetric tensor product
static void	addTensorProduct(Matrix6d D, double s, Matrix3dBase A) Adds the scaled tensor product
static void	addTensorProduct(Matrix6d D, double s, Matrix3dBase A, Matrix3dBase B) Adds the scaled tensor product
static void	addTensorProduct(Matrix6d D, Matrix3dBase A) Adds the tensor product
static void	addTensorProduct4(Matrix6d D, double s, Matrix3dBase A) Adds the scaled symmetric tensor product
static void	createElasticityRotation(Matrix6d T, Matrix3dBase R) Creates a 6x6 rotation matrix for transforming the elasticity tensor From Brown University Notes
static void	createElasticityUnrotation(Matrix6d T, Matrix3dBase R) Creates a 6x6 "unrotation" matrix for transforming the elasticity tensor (i.e.
static void	main(java.lang.String[] args)
static void	rotateTangent(Matrix6d DR, Matrix6d D1, Matrix3dBase R) Rotates a 6x6 material tangent matrix into a new coordinate system.
static void	rotateTangentOld(Matrix6d DR, Matrix6d D1, Matrix3dBase R) Rotates a 6x6 material tangent matrix into a new coordinate system.
static void	unrotateTangent(Matrix6d DR, Matrix6d D1, Matrix3dBase R) Rotates a 6x6 material tangent matrix into a new coordinate system.
static void	v3DotTens4sDotv3(Matrix3d out, Vector3d a, Matrix6d T, Vector3d b) Computes the following formula: v3a .

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

TensorUtils

public TensorUtils()

Method Detail

addSymmetricIdentityProduct

public static void addSymmetricIdentityProduct(Matrix6d D,
                                               Matrix3dBase S)

Adds the scaled tensor product

  S (X) I + I (X) S
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, S is a symmetric 3 x 3 matrix, I is the 3 x 3 identity matrix, and X denotes a tensor product.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

S - matrix for forming the product. This is assumed to be symmetric and only the upper triangular components are used.

addSymmetricTensorProduct

public static void addSymmetricTensorProduct(Matrix6d D,
                                             Matrix3dBase A,
                                             Matrix3dBase B)

Adds the symmetric tensor product

  A (X) B + B (X) A
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, A and B are 3 x 3 matrices (assumed to be symmetric), and (X) denotes a tensor product.

Note that this method sets only the upper triangular components of D.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

A - first matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

B - second matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

addSymmetricTensorProduct

public static void addSymmetricTensorProduct(Matrix6d D,
                                             double s,
                                             Matrix3dBase A,
                                             Matrix3dBase B)

Adds a scaled symmetric tensor product

  s (A (X) B + B (X) A)
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, A and B are 3 x 3 matrices (assumed to be symmetric), and (X) denotes a tensor product.

Note that this method sets only the upper triangular components of D.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scale factor

A - first matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

B - second matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

addTensorProduct

public static void addTensorProduct(Matrix6d D,
                                    Matrix3dBase A)

Adds the tensor product

  A (X) A
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, A is a 3 x 3 matrix (assumed to be symmetric), and (X) denotes a tensor product.

Note that this method sets only the upper triangular components of D.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

A - matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

addTensorProduct

public static void addTensorProduct(Matrix6d D,
                                    double s,
                                    Matrix3dBase A)

Adds the scaled tensor product

  s (A (X) A)
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, A is a 3 x 3 matrix (assumed to be symmetric), and (X) denotes a tensor product.

Note that this method sets only the upper triangular components of D.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scale factor

A - matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

addTensorProduct

public static void addTensorProduct(Matrix6d D,
                                    double s,
                                    Matrix3dBase A,
                                    Matrix3dBase B)

Adds the scaled tensor product

  s (A (X) B)
 

to a 4th order constitutive tensor represented as a 6 x 6 matrix. Here, A and B are 3 x 3 matrices (assumed to be symmetric), and (X) denotes a tensor product.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scale factor

A - matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

B - matrix for forming the product, assumed to be symmetric with only the upper triangular components used.

addScaledIdentityProduct

public static void addScaledIdentityProduct(Matrix6d D,
                                            double s)

Adds the scaled tensor product

  s ( I (X) I )
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, s is a scalar, I is the 3 x 3 identity matrix, and X denotes a tensor product.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scaling factor for the product

addScaledIdentity

public static void addScaledIdentity(Matrix6d D,
                                     double s)

Adds a scaled identity to a 4th order constitutive tensor represented as a 6 x 6 matrix.

Parameters:

s - scaling factor for the identity

addScaled4thPowerProduct

public static void addScaled4thPowerProduct(Matrix6d D,
                                            double s,
                                            Vector3d a)

Adds the scaled tensor product

  s ( a X a X a X a )
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, s is a scalar, a is a 3-vector, and X denotes a tensor product.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scaling factor for the product

a - vector defining the product

addTensorProduct4

public static void addTensorProduct4(Matrix6d D,
                                     double s,
                                     Matrix3dBase A)

Adds the scaled symmetric tensor product

  s (A_ik A_jl + A_il A_jk)/2 
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, s is a scalar and A is a 3 x 3 matrix (assumed to be symmetric).

Note that this method sets only the upper triangular components of D.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scaling factor for the product

A - matrix used to form the product (assumed symmetric with only the upper triangular part used)

addSymmetricTensorProduct4

public static void addSymmetricTensorProduct4(Matrix6d D,
                                              double s,
                                              Matrix3dBase A,
                                              Matrix3dBase B)

Adds the scaled symmetric tensor product

 s (A_ik B_jl + A_il B_jk)/2 + (B_ik A_jl + B_il A_jk)/2
 

to a symmetric 4th order constitutive tensor represented as a 6 x 6 matrix. Here, s is a scalar and A is a 3 x 3 matrix (assumed to be symmetric).

Note that this method sets only the upper triangular components of D.

Parameters:

D - 6 x 6 matrix representation of the tensor to which the product is added

s - scaling factor for the product

A - first matrix in the product (assumed symmetric with only the upper triangular part used)

B - second matrix in the product (assumed symmetric with only the upper triangular part used)

rotateTangentOld

public static void rotateTangentOld(Matrix6d DR,
                                    Matrix6d D1,
                                    Matrix3dBase R)

Rotates a 6x6 material tangent matrix into a new coordinate system. The rotation is specified by a rotation matrix R that transforms from the new coordinate system into the current one. The transformation is based on the general 4th order elasticity tensor rotation formula

A'_ijkl = sum R_ai R_bj R_ck R_dl A_abcd

but is optimized and also accomodates the fact that the 6x6 tangent matrix is a reduced version of the general elasticity tensor. It also assumes that the elasticity tensor is symmetric. Each entry in the tangent matrix corresponds to ijkl indices of the elasticity tensor as follows:

[ 0000 0011 0022 0001 0012 0002 1100 1111 1122 1101 1112 1102 2200 2211 2222 2201 2212 2202 0100 0111 0122 0101 0112 0102 1200 1211 1222 1201 1212 1202 0200 0211 0222 0201 0212 0202 ]

Note that this indexing convention follows that used in FEBio but is not standard; in particular, the lower and right entries are sometimes interchanged depending on how the stress/strain tensors are vectorized.

Parameters:

DR - result is returned here

D1 - original tangent matrix to rotate

R - rotation matrix

createElasticityRotation

public static void createElasticityRotation(Matrix6d T,
                                            Matrix3dBase R)

Creates a 6x6 rotation matrix for transforming the elasticity tensor From Brown University Notes

Parameters:

T - output 6x6 matrix

R - 3D rotation matrix

createElasticityUnrotation

public static void createElasticityUnrotation(Matrix6d T,
                                              Matrix3dBase R)

Creates a 6x6 "unrotation" matrix for transforming the elasticity tensor (i.e. uses R') From Brown University Notes

Parameters:

T - output 6x6 matrix

R - 3D rotation matrix

rotateTangent

public static void rotateTangent(Matrix6d DR,
                                 Matrix6d D1,
                                 Matrix3dBase R)

Rotates a 6x6 material tangent matrix into a new coordinate system. The rotation is specified by a rotation matrix R that transforms from the new coordinate system into the current one.

Parameters:

DR - result is returned here

D1 - original tangent matrix to rotate

R - rotation matrix

unrotateTangent

public static void unrotateTangent(Matrix6d DR,
                                   Matrix6d D1,
                                   Matrix3dBase R)

Rotates a 6x6 material tangent matrix into a new coordinate system. The rotation is specified by the transpose of rotation matrix R

Parameters:

DR - result is returned here

D1 - original tangent matrix to rotate

R - rotation to "unrotate"

main

public static void main(java.lang.String[] args)

v3DotTens4sDotv3

public static void v3DotTens4sDotv3(Matrix3d out,
                                    Vector3d a,
                                    Matrix6d T,
                                    Vector3d b)

Computes the following formula: v3a . T4S . v3b where v3a and v3b are both vector-3, T4S is a symmetrical 4th order tensor, and . is the dot product symbol. FEBio: tens4ds::vdotTdotv

Parameters:

out - Result to be stored in. 3x3 matrix.

a - Left vector3

T - Symmetrical 4th order tensor. E.g. material stress matrix.

b - Right vector3