Local coordinate systems of 3D elements

X.B.4.5 Local coordinate systems of 3D elements

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: $r_{1} = R$ , $r_{2} = S$ , $r_{3} = T$ . So the element coordinate system is ${O, e_{i}}$ and the three vectors R, S and T are denoted $r_{i}$ . Then one defines with matrix $A$ corresponding to the description of vectors $r_{i}$ on the base $e_{i}$ . One has:

r_{i} = A_{i j} e_{j},

A_{i j} = r_{i} \cdot e_{j} .

The above statement is equivalent to state that matrix $A_{i j}$ must be symmetric. So the problem reduces to “find three base vectors $e_{i}$ such that:

r_{i} \cdot e_{j} = r_{j} \cdot e_{i} .

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 $e_{i}$ and the three vectors $r_{i}$ provided as data, one must find the three components of the vector $w = θ v$ such that the new base vector $e_{i}^{'}$ obtained with equation (X.B.1) satisfy the relation

r_{i} \cdot e_{j}^{'} - r_{j} \cdot e_{i}^{'} = 0 .

(X.B.2)

This set of equations is non linear in the primary unknowns $w_{i}$ and one solves them by a Newton-Raphson in which successive approximations of the three vectors $e_{i}$ are calculated. One performs a Taylor expansion of the relation (X.B.2) stopped at the first order:

e_{j}^{'} \approx e_{j} + Δ e_{j},

Δ e_{j} \approx Δ w \times e_{j} .

Substituting the two previous expressions in (X.B.2), one obtains successively:

r_{i} \cdot (e_{j} + Δ e_{j}) - r_{j} \cdot (e_{i} + Δ e_{i}) = 0,

r_{i} \cdot (e_{j} + Δ w \times e_{j}) - r_{j} \cdot (e_{i} + Δ w \times e_{i}) = 0,

r_{i} \cdot (Δ w \times e_{j}) - r_{j} \cdot (Δ w \times e_{i}) = r_{j} \cdot e_{i} - r_{i} \cdot e_{j},

(e_{j} \times r_{i} - e_{i} \times r_{j}) \cdot Δ w = r_{j} \cdot e_{i} - r_{i} \cdot e_{j} .

(X.B.3)

This last expression allows to derive three independent linear equations, with three unknowns $w_{i}$ . The Newton-Raphson algorithm used looks as follows:

1.

Determine a first estimate of the three vectors

e_{i}

. In FeResPost, one chooses

e_{1}

parallel to R,

e_{2}

perpendicular to plane defined by vectors R and T, and

e_{3}

perpendicular to

e_{1}

and

e_{2}

2.

Then one iterates until convergence:

(a)

Vector

Δ w

is obtained by solving (X.B.3).

(b)

One then obtains the rotation angle

θ

and unit rotation vector

v

θ = ∥ Δ w ∥,

v = \frac{Δ w}{∥ Δ w ∥} .

(c)

New estimates of the base vectors are obtained using (X.B.1).

One assumes convergence if $θ < 1 0^{- 5}$ .

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