The modification of coordinate system into which the components of vectorial or tensorial results are expressed is a common operation performed when post-processing finite element results. To perform the transformation some vectorial and tensorial calculation is necessary. One gives in this Appendix, a summary of theoretical background necessary to understand the operations performed with FeResPost.
When performing tensorial calculations, or post-processing results, one manipulates 1D or 2D arrays of real values corresponding to the components of vectors or tensors in a specified coordinate system. Higher order tensors also exist, but they are not manipulated in FeResPost and we do not present the theory for tensor order larger than 2.
A vectorial force can be expressed by its components in the Cartesian coordinate system characterized by its origin and its three unit length mutually orthogonal vectors , and . Then the vector corresponds to:
In the rest of the text the components of a vector are denoted and one uses the Bose-Einstein convention of summation on repeated indices so that the previous expression is simply written:
A Cauchy stress tensor is characterized by its components in the Cartesian coordinate system . (One uses the same notations as in section X.B.1.1).
The nine quantities can be considered as the basic tensors from which all the other tensors are obtained linear combinations. Note that the Cauchy stress tensor is always symmetrical so that .
In sections X.B.1.1 and X.B.1.2, one presented vector and tensor components in Cartesian coordinate systems. The same definition is also worthy in curvilinear coordinate systems. However, the director vectors depend on the point on which the vector or tensor is attached. Conventionally, one decides that the director vectors are chosen tangent to the coordinate lines and are of unit length.
For example, for a cylindrical coordinate system, the position of a point depends on three coordinates , and . So one has:
Then three tangent vectors are obtained by deriving the position wrt coordinates:
This process to define base vectors at any point can be generalized to all curvilinear coordinate systems. However, the cylindrical and spherical coordinate systems have a peculiarity: at a given point, the three base vectors are mutually orthogonal. This is not a general characteristic of curvilinear coordinate systems.
The orthogonality property of the coordinate systems one uses in FeResPost simplifies the transformation of components from one coordinate system to another. Indeed such transformations reduce to transformations between Cartesian coordinate systems. There is only one difficulty in this process: to calculate the base vectors at every point.
One considers only the peculiar case of two Cartesian coordinate systems with the same origin. In section X.B.1.3, one showed that even for cylindrical and spherical coordinate systems it is possible to reduce the complexity of the problem to a transformation of the components between two Cartesian coordinate systems.
One considers a vectors with components expressed in two Cartesian coordinate systems with base vectors and respectively. So one has:
In the last expression we introduce the notations and for the components of vector in and respectively.
It is possible to decompose the vectors as a linear combination of vectors :
The coefficients are easily calculated. Indeed, the scalar multiplication of the previous equality by gives successively:
By a similar calculation, it is possible to identify the relations between the components of vector expressed in the two coordinate systems:
That's the reason why the transformation is called a covariant transformation. One also says that or are covariant components of vector in coordinate systems and respectively. In the last vector component transformation one recognizes a classical algebraic result:
The matrix is orthogonal: and the reverse relation for vector components is:
Similarly to what has been done for vectors in section X.B.2.1, one derives transformations of the components of tensors. One considers the components of tensor in two coordinate systems:
Using the same definition of transformation matrix as in section X.B.2.1, one writes:
Then, the substitution of the two expressions in the equations defining components gives:
Here again, one recognizes a classical matricial expression:
A transformation of coordinate systems commonly done is a rotation of the coordinate system around a specified axis passing through the origin of the coordinate system. The resulting coordinate system has the same origin, but its base vectors are modified. A classical use of this operation corresponds to the transformation of vector or tensor results in material, ply or element axes of 2D elements.
For example, let be a unit vector defining the rotation axis and the rotation angle. Then the three transformed base vectors are given by:
One gives here additional information on the modification of coordinate systems with Result method ``modifyRefCoordSys''. When developing the various transformations possible with this function, one tried to keep as much as possible, the correspondence with Patran (Patran 90) transformations. Sometimes, this process has been partially done by trials and guesses until an agreement was found.
In the rest of the section, one gives the information allowing the user to determine exactly the operations that are performed on finite element entities and coordinate systems to defined the transformed coordinate system. One makes the distinction between local, global and projected coordinate systems.
Several types of local coordinate systems may be defined, and the operations performed to define the coordinate system depend on the case:
Global coordinate systems correspond to CoordSys objects. However one makes the distinction between:
The projected coordinate system is a local Cartesian coordinate system the definition of which depends on the type of element:
For CQUAD4 Nastran elements, the origin of the element coordinate system is defined to be the intersection of straight lines AC and BD( A, B, C, and D being the corners of the element). As long as the four defining nodes are co-planar, this definition is sufficient. But otherwise, the two straight lines do not intersection, and a generalization of the definition of the origin has to be found. We decide that the origin of the coordinate system shall be the point closest to the two straight lines AC and BD.
The two straight lines can be defined with corresponding parameterized equations:
So, one has to find the parameters and that minimize . The vector can be developed as follows:
The square of the norm defined above depends on parameters and and is given by:
This function must be stationary at the optimum point. Therefore its first derivatives wrt and must be zero:
This leads to a system of two linear equations with the two unknowns and :
These two equations may be rewritten as follows:
This equation simply means that the vector connecting the two optimum points is perpendicular to both lines AC and BD. Finally, after resolution of the system of equations and various substitutions, on finds the origin of coordinate system at:
In Nastran, for 3D elements, the definition of the local element coordinate system is a little tricky, and it is not easy to interpret the information found in the reference manuals. One provides here the interpretation that has been used to build the coordinate systems of 3D elements in FeResPost.
The first step of the local element construction is to build three vectors R, S and T related to the geometry of the element. The way those three vectors are constructed depends on the 3D element that is being constructed. Then the R, S and T vectors are used to build a local Cartesian coordinate system. In Nastran Quick Reference Guide [Sof04b], on gives the following explanation for the CTETRA element:
The element coordinate system is chosen as close as possible to the R, S, and T vectors and points in the same general direction. (Mathematically speaking, the coordinate system is computed in such a way that, if the R, S, and T vectors are described in the element coordinate system, a 3x3 positive definite symmetric matrix would be produced.)In FeResPost, one makes the assumption that this information is also true for the other 3D elements CHEXA and CPENTA. One gives here the mathematical development that leads us to the definition of local coordinate system.
First, let us introduce the notations: , , . So the element coordinate system is and the three vectors R, S and T are denoted . Then one defines with matrix corresponding to the description of vectors on the base . One has:
The above statement is equivalent to state that matrix must be symmetric. So the problem reduces to ``find three base vectors such that:
Note also that this condition makes us think to the polar decomposition theorem that states that a positive definite tensor can be decomposed in the product of an orthogonal tensor and a pure symmetric positive definite tensor. This means that any deformation of a continuum medium can be decomposed in a rigid rotation and a pure deformation. One is actually interested in the rigid rotation that can be expressed by its rotation vector (see section X.B.3). So considering an initial set of base vectors and the three vectors provided as data, one must find the three components of the vector such that the new base vector obtained with equation (X.B.1) satisfy the relation
Substituting the two previous expressions in (X.B.2), one obtains successively:
Vectorial and tensorial components of Results are sometimes expressed in coordinate systems related to material properties. Of course, this is also the case when laminated properties are attributed to the elements. Then, three different coordinate systems can be used: ``plyCS'', ``lamCS'' and ``matCS''.
When vectorial or tensorial Results on composite elements are imported, they are expressed:
When one modifies the reference coordinate system of a Result, the components are actually modified according to the rules presented in Table X.B.1.