[cig-commits] r12640 - doc/cigma/manual

[sue at geodynamics.org]

Author: sue
Date: 2008-08-14 17:16:17 -0700 (Thu, 14 Aug 2008)
New Revision: 12640

Added:
   doc/cigma/manual/workfile1.lyx
Log:
content and equations from linear tetrahedron

Added: doc/cigma/manual/workfile1.lyx
===================================================================
--- doc/cigma/manual/workfile1.lyx	                        (rev 0)
+++ doc/cigma/manual/workfile1.lyx	2008-08-15 00:16:17 UTC (rev 12640)
@@ -0,0 +1,647 @@
+#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\usepackage{hyperref}
+
+\let\myUrl\url
+\renewcommand{\url}[1]{(\myUrl{#1})}
+
+\newcommand{\mynote}[1]{}
+\end_preamble
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry true
+\use_amsmath 1
+\use_esint 0
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\leftmargin 1in
+\topmargin 1in
+\rightmargin 1in
+\bottommargin 1in
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language swedish
+\papercolumns 1
+\papersides 2
+\paperpagestyle headings
+\tracking_changes false
+\output_changes false
+\author "" 
+\author "" 
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Cigma
+\end_layout
+
+\begin_layout Part
+Workfile 1
+\end_layout
+
+\begin_layout Chapter
+The Linear Tetrahedron
+\end_layout
+
+\begin_layout Section
+15.2 The Linear Tetrahedron
+\end_layout
+
+\begin_layout Standard
+The linear tetrahedron, shown in figure 15.1(a), is not used often for stress
+ analysis because of its poor performance.
+\begin_inset Foot
+status open
+
+\begin_layout Standard
+Derivative of shape functions are constant over the element volume.
+ Strains and stresses recovered in this manner can be highly inaccurate.
+ This makes the element dangerous for stress analysis.
+ On the other hand, when the objective is merely to get values of primary
+ variables, as in thermal analysis and computational gas dynamics, the linear
+ tetrahedron is acceptable.
+\end_layout
+
+\end_inset
+
+ Its main value in structural and solid mechanics is educational: it serves
+ as a vehicle to introduce the basic steps of formulation of 3D solid elements,
+ particularly as regards use of natural coordinate systems and node numbering
+ conventions.
+ It should be noted that 3D visualization is notoriously more difficult
+ than 2D, so we need to proceed somewhat slowly here.
+\end_layout
+
+\begin_layout Standard
+\align center
+Figure 15.1a and 15.1b
+\end_layout
+
+\begin_layout Subsection
+15.2.1 Tetrahedron Geometry
+\end_layout
+
+\begin_layout Standard
+Figure 15.1 shows a typical 4-node tetrahedron.
+ Its geometry is fully defined by giving the location of the four corner
+ nodes with respect to the global RCC system (
+\emph on
+x, y, z
+\emph default
+):
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+x_{i},\, y_{i},\, z_{i}\,\left(i=1,\,2,\,3,\,4\right)\label{eq:1}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The volume measure of the tetrahedron is denoted
+\begin_inset Foot
+status open
+
+\begin_layout Standard
+This symbol (Upsilon) is used to avoid confusion with 
+\emph on
+V
+\emph default
+, which denotes the volume of a generic body.
+\end_layout
+
+\end_inset
+
+ by 
+\begin_inset Formula ${\normalcolor \nu}$
+\end_inset
+
+ and is given by the following determinant:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\nu=\frac{1}{6}\mathrm{det}\left[\begin{array}{cccc}
+1 & 1 & 1 & 1\\
+x_{1} & x_{2} & x_{3} & x_{4}\\
+y_{1} & y_{2} & y_{3} & y_{4}\\
+z_{1} & z_{2} & z_{3} & z_{4}\end{array}\right]\label{eq:2}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This volume is a 
+\emph on
+signed
+\emph default
+ quantity.
+ It is positive if the corners are numbered in such a way that the volume
+ is positive.
+ A numbering rule that achieves this goal is as follows:
+\end_layout
+
+\begin_layout Enumerate
+Pick a corner as initial one.
+ In Fig.
+ 15.1(a) this is numbered 1.
+\end_layout
+
+\begin_layout Enumerate
+Pick a face that will contain the first three corners.
+ The excluded corner will be the last one.
+\end_layout
+
+\begin_layout Enumerate
+Number these three corners in a counterclockwise sense when looking at the
+ face from the excluded corner.
+ See Fig.
+ 15.1(b).
+\end_layout
+
+\begin_layout Standard
+In what follows, we shall always assume that the numbering has been done
+ in that manner so that 
+\begin_inset Formula $\nu>0$
+\end_inset
+
+.
+\begin_inset Foot
+status open
+
+\begin_layout Standard
+The tetrahedron volume can be zero only if the four corners are coplanar.
+ This case will be excluded.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\align center
+Figure 15.2a and 15.2b
+\end_layout
+
+\begin_layout Subsection
+15.2.2 Tetrahedral Coordinates
+\end_layout
+
+\begin_layout Standard
+The set of tetrahedral coordinates 
+\begin_inset Formula $\zeta_{1},$
+\end_inset
+
+ 
+\begin_inset Formula $\zeta_{2},$
+\end_inset
+
+ 
+\begin_inset Formula $\zeta_{3},$
+\end_inset
+
+ 
+\begin_inset Formula $\zeta_{4}$
+\end_inset
+
+ is the three-dimensional analog of the triangular coordinate set discussed
+ in Chapter 15 of IFEM.
+ The value of 
+\begin_inset Formula $\zeta_{1}$
+\end_inset
+
+ is one at corner 
+\emph on
+
+\begin_inset Formula $i$
+\end_inset
+
+
+\emph default
+, zero at the other 3 corners (i.e., on the opposite face) and varies linearly
+ as one traverses the distance from the corner to the face.
+ The sum of the four coordinates is identically one:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\zeta_{1}+\zeta_{2}+\zeta_{3}+\zeta_{4}=1\label{eq:3}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Any function linear in 
+\begin_inset Formula $x,\, y,\, z$
+\end_inset
+
+, say 
+\begin_inset Formula $F\left(x,\, y,\, z\right)$
+\end_inset
+
+, that takes the values 
+\begin_inset Formula $F_{1}\left(i=1,2,3,4\right)$
+\end_inset
+
+ at the corners may be interpolated in terms of the tetrahedron coordinates
+ as
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+F\left(\zeta_{1}+\zeta_{2}+\zeta_{3}+\zeta_{4}\right)=F_{1}\zeta_{1}+F_{2}\zeta_{2}+F_{3}\zeta_{3}+F_{4}\zeta_{4}=F_{i}\zeta_{i}.\label{eq:4}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Example 15.1.
+
+\series default
+ Suppose that 
+\begin_inset Formula $F\left(x,\, y,\, z\right)=4x+9y-8z+3$
+\end_inset
+
+ and that the coordinates of corners 1,2,3,4 are (0,0,0), (1,0,0), (0,1,0),
+ and (0,0,1) respectively.
+ The values of F at the corners are 
+\begin_inset Formula $F_{1}=3,\, F_{2}=7,\, F_{3}=12$
+\end_inset
+
+ and 
+\begin_inset Formula $F_{4}=-5$
+\end_inset
+
+.
+ Consequently, 
+\begin_inset Formula $F\left(\zeta_{1}+\zeta_{2}+\zeta_{3}+\zeta_{4}\right)=3\zeta_{1}+7\zeta_{2}+12\zeta_{3}-5\zeta_{4}.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+15.2.3 Coordinate Transformations
+\end_layout
+
+\begin_layout Standard
+The geometric definition of the element in terms of these coordinates is
+ obtained by applying the geometry definition (Eq.
+ 
+\begin_inset LatexCommand ref
+reference "eq:4"
+
+\end_inset
+
+) to 
+\emph on
+x, y, 
+\emph default
+and 
+\emph on
+z, 
+\emph default
+and appending the sum-of-coordinates constraint (Eq.
+ 
+\begin_inset LatexCommand ref
+reference "eq:3"
+
+\end_inset
+
+):
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\left[\begin{array}{c}
+1\\
+x\\
+y\\
+z\end{array}\right]=\left[\begin{array}{cccc}
+1 & 1 & 1 & 1\\
+x_{1} & x_{2} & x_{3} & x_{4}\\
+y_{1} & y_{2} & y_{3} & y_{4}\\
+z_{1} & z_{2} & z_{3} & z_{4}\end{array}\right]\left[\begin{array}{c}
+\zeta_{1}\\
+\zeta_{2}\\
+\zeta_{3}\\
+\zeta_{4}\end{array}\right].\label{eq:5}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Inverting this relation gives
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\left[\begin{array}{c}
+\zeta_{1}\\
+\zeta_{2}\\
+\zeta_{3}\\
+\zeta_{4}\end{array}\right]=\frac{1}{6\nu}\left[\begin{array}{cccc}
+6\nu_{1} & a_{1} & b_{1} & c_{1}\\
+6\nu_{2} & a_{2} & b_{2} & c_{2}\\
+6\nu_{3} & a_{3} & b_{3} & c_{3}\\
+6\nu_{4} & a_{4} & b_{4} & c_{4}\end{array}\right]\left[\begin{array}{c}
+1\\
+x\\
+y\\
+z\end{array}\right]\label{eq:6}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+where the coefficients of this matrix can be calculated by forming the adjoints
+ of the matrix in Eq.
+ 
+\begin_inset LatexCommand ref
+reference "eq:5"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Remark 15.1 
+\series default
+The values of 
+\begin_inset Formula $a_{i}$
+\end_inset
+
+, 
+\begin_inset Formula $b_{i}$
+\end_inset
+
+, and 
+\begin_inset Formula $c_{i}$
+\end_inset
+
+ obtained by explicit inversion are
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\begin{array}{ccc}
+a_{1}=y_{2}z_{43}-y_{3}z_{42}+y_{4}z_{32}, & b_{1}=x_{2}z_{43}-x_{3}z_{42}+x_{4}z_{32}, & c_{1}=x_{2}y_{43}-x_{3}y_{42}+x_{4}y_{32},\\
+a_{2}=y_{1}z_{43}-y_{3}z_{41}+y_{4}z_{31}, & b_{2}=x_{1}z_{43}-x_{3}z_{41}+x_{4}z_{31}, & c_{2}=x_{1}y_{43}-x_{3}y_{41}+x_{4}y_{31},\\
+a_{3}=y_{1}z_{42}-y_{2}z_{41}+y_{4}z_{21}, & b_{3}=x_{1}z_{42}-x_{2}z_{41}+x_{4}z_{21}, & c_{3}=x_{1}y_{42}-x_{2}y_{41}+x_{4}y_{21},\\
+a_{4}=y_{1}z_{32}-y_{2}z_{31}+y_{3}z_{21}, & b_{4}=x_{1}z_{32}-x_{2}z_{31}+x_{3}z_{21}, & c_{4}=x_{1}y_{32}-x_{2}y_{31}+x_{3}y_{21},\end{array}\label{eq:7}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which the abbreviations 
+\begin_inset Formula $x_{ij}=x_{i}-x_{j},\, y_{ij}=y_{i}-y_{j}$
+\end_inset
+
+ and 
+\begin_inset Formula $z_{ij}=z_{i}-z_{j}$
+\end_inset
+
+ are used.
+ The volume is given explicitly by
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+6\nu=x_{21}\left(y_{31}z_{41}-y_{41}z_{31}\right)+y_{21}\left(x_{41}z_{31}-x_{31}z_{41}\right)+z_{21}\left(x_{31}y_{41}-x_{41}y_{31}\right).\label{eq:8}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The values of 
+\begin_inset Formula $\nu_{1}$
+\end_inset
+
+ are of no interest in what follows.
+\end_layout
+
+\begin_layout Subsection
+15.2.4 Geometric Interpretation
+\end_layout
+
+\begin_layout Standard
+Figure 15.2 illustrates two geometric interpretations of coordinate 
+\begin_inset Formula $\zeta_{1}.$
+\end_inset
+
+ In Fig.
+ 15.2(a), 
+\begin_inset Formula $\zeta_{1}=C$
+\end_inset
+
+, where 
+\begin_inset Formula $C$
+\end_inset
+
+ is a number between 0 and 1, is the equation of a plane parallel to the
+ face 234.
+ The plane coincides with that face is 
+\begin_inset Formula $\zeta_{1}=0$
+\end_inset
+
+, it passes through corner node 1 if 
+\begin_inset Formula $\zeta_{1}=1$
+\end_inset
+
+, and is interpolated linearly in between.
+\end_layout
+
+\begin_layout Standard
+Fig.
+ 15.2(b) illustrates another interpretation that appears in many FEM books.
+ Consider a point 
+\begin_inset Formula $P$
+\end_inset
+
+ of coordinates 
+\begin_inset Formula $\left(\zeta_{1},\,\zeta_{2},\,\zeta_{3},\,\zeta_{4}\right)$
+\end_inset
+
+ inside the tetrahedron.
+ Joining 
+\begin_inset Formula $P$
+\end_inset
+
+ to the corners we obtain four sub-tetrahedra 234
+\begin_inset Formula $P$
+\end_inset
+
+, 341
+\begin_inset Formula $P$
+\end_inset
+
+, 412
+\begin_inset Formula $P$
+\end_inset
+
+, and 123
+\begin_inset Formula $P$
+\end_inset
+
+, whose volumes are 
+\begin_inset Formula $\nu_{1},\,\nu_{2},\,\nu_{3},$
+\end_inset
+
+ and 
+\begin_inset Formula $\nu_{4}$
+\end_inset
+
+, respectively.
+ Then 
+\begin_inset Formula $\zeta_{1}$
+\end_inset
+
+ is the ratio 
+\begin_inset Formula $\nu_{1}/\nu$
+\end_inset
+
+.
+ Fig.
+ 15.2(b) pictures the sub-tetrahetron 234
+\begin_inset Formula $P$
+\end_inset
+
+ of volume 
+\begin_inset Formula $\nu_{1}$
+\end_inset
+
+.
+ On account of this relation, tetrahedral coordinates are also called volume
+ coordinates.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Remark 15.2
+\series default
+ The interpretation as volume coordinates only holds for the tetrahedron
+ defined by 4 corner nodes.
+ It fails for higher order tetrahedra defined by additional nodes (e.g., midpoints
+).
+ For this reason, the second interpretation, as well as the name ``volume
+ coordinates,'' will not be used here.
+\end_layout
+
+\begin_layout Subsection
+15.2.5 Partial Derivatives
+\end_layout
+
+\begin_layout Standard
+From Equations 
+\begin_inset LatexCommand ref
+reference "eq:5"
+
+\end_inset
+
+ and 
+\begin_inset LatexCommand ref
+reference "eq:6"
+
+\end_inset
+
+, we can easily find the following relations for the partial derivatives
+ of Cartesian and tetrahedral coordinates
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\begin{array}{ccc}
+\frac{\partial x}{\partial\zeta_{i}}=x_{i} & \frac{\partial y}{\partial\zeta_{i}}=y_{i} & \frac{\partial z}{\partial\zeta_{i}}=z_{i}\end{array}\label{eq:9}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+6\nu\frac{\partial\zeta_{i}}{\partial x}=a_{i},\,\,\,\frac{\partial\zeta_{i}}{\partial y}=b_{i},\,\,\,\frac{\partial\zeta_{i}}{\partial z}=c_{i}\label{eq:10}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The derivative of a function 
+\begin_inset Formula $F\left(\zeta_{1},\,\zeta_{2},\,\zeta_{3},\,\zeta_{4}\right)$
+\end_inset
+
+with respect to the Cartesian coordinates follows from Eq.
+ 
+\begin_inset LatexCommand ref
+reference "eq:10"
+
+\end_inset
+
+ and the chain rule:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\begin{array}{c}
+\frac{\partial F}{\partial x}=\frac{\partial F}{\partial\zeta_{i}}\frac{\partial\zeta_{i}}{\partial x}=\frac{1}{6\nu}\left(\frac{\partial F}{\partial\zeta_{1}}a_{1}+\frac{\partial F}{\partial\zeta_{2}}a_{2}+\frac{\partial F}{\partial\zeta_{3}}a_{3}+\frac{\partial F}{\partial\zeta_{4}}a_{4}\right)=\frac{1}{6\nu}\frac{\partial F}{\partial\zeta_{i}}a_{i}.\\
+\frac{\partial F}{\partial y}=\frac{\partial F}{\partial\zeta_{i}}\frac{\partial\zeta_{i}}{\partial y}=\frac{1}{6\nu}\left(\frac{\partial F}{\partial\zeta_{1}}b_{1}+\frac{\partial F}{\partial\zeta_{2}}b_{2}+\frac{\partial F}{\partial\zeta_{3}}b_{3}+\frac{\partial F}{\partial\zeta_{4}}b_{4}\right)=\frac{1}{6\nu}\frac{\partial F}{\partial\zeta_{i}}b_{i}.\\
+\frac{\partial F}{\partial z}=\frac{\partial F}{\partial\zeta_{i}}\frac{\partial\zeta_{i}}{\partial z}=\frac{1}{6\nu}\left(\frac{\partial F}{\partial\zeta_{1}}c_{1}+\frac{\partial F}{\partial\zeta_{2}}c_{2}+\frac{\partial F}{\partial\zeta_{3}}c_{3}+\frac{\partial F}{\partial\zeta_{4}}c_{4}\right)=\frac{1}{6\nu}\frac{\partial F}{\partial\zeta_{i}}c_{i}.\end{array}\label{eq:11}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\end_body
+\end_document