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--- playground:tempjim [2014/08/27 17:58] – jim_brouzoulis
+++ playground:tempjim [2014/08/28 21:53] (current) – jim_brouzoulis
@@ Line 1: / Line 1: @@
+\(
+   \renewcommand{\b}[1]{\mathbf{#1}}
+\)
+$$\b{a}_{\rm{i}} \rm{b} b$$
 ====== Implementation of a linear plane stress triangle ======
 This tutorial will describe the implementation of a standard plane stress triangle with linear approximation of the displacement field.
+For quasi-static problems the following set of equations needs to be solved
+$$\mathbf{f}_{\mathrm{int}} = \mathbf{f}_{\mathrm{ext}}$$
+with the internal and external force vectors respectively
+$$\b{f}_{\rm{int}} = \int_V \b{B}^{\rm{T}} \b{\sigma} \ \rm{d}V $$
-Each element, in the structural module (SM) needs to compute the following quantities:
+$$\b{f}_{\rm{ext}} = \int_V \b{N}^{\rm{T}} \b{b} \ \rm{d}V
++ \int_{\Gamma} \b{N}^{\rm{T}} \b{t} \ \rm{d}\Gamma $$
-Internal load vector
+The default solution procedure for solving the equations are a Newton-Rapshon scheme and for this the tangent stiffness matrix is needed,
-$$\mathbf{f}_{\mathrm{int}} = \int_V \mathbf{B}^{\mathrm{T}} \mathbf{\sigma} \ \mathrm{d}V $$
+computed as
-External load vector
-$$\mathbf{f}_{\mathrm{ext}} = \int_V \mathbf{N}^{\mathrm{T}} \mathbf{b} \ \mathrm{d}V
-+ \int_{\Gamma} \mathbf{N}^{\mathrm{T}} \mathbf{t} \ \mathrm{d}\Gamma $$
-Tangent stiffness matrix
 $$\mathbf{K} = \int_V \mathbf{B}^{\mathrm{T}} \mathbf{D} \mathbf{B} \ \mathrm{d}V $$
+Since the FE equations for all standard continuum elements (2D plane stress/strain, 3D, etc.) are equal, much
-Since the FE equations for all standard continuum elements (2D plane stress/strain, 3D, etc.) are equal much
+of the element implementations are placed in the base class ''NonLinStructuralElement'' (or ''StructuralElement'' for purely linear problems).
-of the element implementations are placed in the base class ''NonLinStructuralElement''.
+For example, this base class implements methods to compute the internal and external load vector together with the tangent stiffnes matrix (all as defined above).
+The main question that arises is then: what must a new element implement? In short it is everything that is element specific, such as
-This class implements the integration over the volume and the boundary. What the element needs to implement or specify is:
+the following:
   * \(\mathbf{N}\) and \(\mathbf{B}\) matrices
-  * An integration rule (number of integration points etc.)
+    * These depends on the number of nodes and the number o dofs stored in each node.
+  * An integration rule
+    * This defines what type of integration alogorithm (Gauss, Newton-Cotes, Lobatto), the number of integration points,
+    * if there should be several algorithms to support reduced integration for example.
   * Compute the differential volume element \(\Delta V \) and \(\Delta \Gamma \)
@@ Line 66: / Line 73: @@
-Computation of the \(\mathbf{B}\) matrix:
+Computation of the \(\mathbf{B}\)-matrix:
 This method is called ''computeBmatrixAt''
-The derivatives of the shape functions is evaluated using the elements' interpolator
+The derivatives of the shape functions ''dNdx'' are evaluated using the elements' interpolator class
 <code c++>
 this->interp.evaldNdx( dNdx, * gp->giveNaturalCoordinates(), FEIElementGeometryWrapper(this) );
@@ Line 82: / Line 89: @@
 $$
+From these derivatives the \(\mathbf{B}\)-matrix can be constructed which for a three noded triangle with two dofs per node looks like
+$$\mathbf{B}
+= \begin{pmatrix}
+\frac{\mathrm{d}N_1}{\mathrm{d}x_1} & 0 & \frac{\mathrm{d}N_2}{\mathrm{d}x_1} & 0 & \frac{\mathrm{d}N_3}{\mathrm{d}x_1} & 0 \\
+\frac{\mathrm{d}N_1}{\mathrm{d}x_1} & 0 & \frac{\mathrm{d}N_2}{\mathrm{d}x_1} & 0 & \frac{\mathrm{d}N_3}{\mathrm{d}x_1} & 0\\
+\frac{\mathrm{d}N_1}{\mathrm{d}x_2} & \frac{\mathrm{d}N_1}{\mathrm{d}x_1} & \frac{\mathrm{d}N_2}{\mathrm{d}x_2}
+& \frac{\mathrm{d}N_2}{\mathrm{d}x_1} & \frac{\mathrm{d}N_3}{\mathrm{d}x_2} & \frac{\mathrm{d}N_3}{\mathrm{d}x_1} \\
+\end{pmatrix}
+$$
 <code c++>
-void
-BasicElement :: computeBmatrixAt(GaussPoint *gp, FloatMatrix &answer, int li, int ui)
-{
-    /* Compute the [3x6] strain-displacement matrix {B} for the element,
-     * evaluated at the given gp.
-     * {B}*{a} should provide the strains {eps} in Voigt form
-     * {eps} = {eps_xx, eps_yy, gam_xy}^T with {a} being the
-     * solution vector of the element.
-     */
-    /* Evaluate the derivatives of the shape functions at the position of the gp.
-     * dNdx = [dN1/dx1 dN1/dx2
-     *         dN2/dx1 dN2/dx2
-     *         dN3/dx1 dN3/dx2]
-    */
-    FloatMatrix dNdx;
-    this->interp.evaldNdx( dNdx, * gp->giveNaturalCoordinates(), FEIElementGeometryWrapper(this) );
     // Construct the B-matrix
     answer.resize(3, 6);
@@ Line 118: / Line 116: @@
     answer.at(3, 5) = dNdx.at(3, 2);
     answer.at(3, 6) = dNdx.at(3, 1);
-}
 </code>
-Part of header file...
+Auxilary metods that needs to be overloaded:
 <code c++>
@@ Line 142: / Line 142: @@
     virtual int testElementExtension(ElementExtension ext) { return ( ( ext == Element_EdgeLoadSupport ) ? 1 : 0 ); }
 </code>
+<file - BasicElement.C>
+</file>
+<file - BasicElement.h>
+</file>
+===== Problem representation - Engineering model =====
+The concept "engineering model" is an abstraction for the problem under
+consideration. It represents the type of analysis to be performed (e.g. static structural, transient heat flow, etc.).
+The base class ''EngngModel'' declares and implements the basic services for assembling
+characteristic components and services for starting the solution step and
+its termination. Derived classes ``know'' the form of governing
+equation and the physical meaning of  particular components.
+They are responsible for forming the governing equation for each solution
+step,  usually by summing contributions from particular elements and
+nodes.ecific load type dependent
+services and implement all necessary services.