Bug/mortar discretization (#88)

* new mortar segmentation tests which are failing

* test_problems_mortar_3d.jl: first test (Tet4) pass

* solvers.jl: diagonal of A is now properly filled, if that option is used. Another option is to remove zero rows from matrix system, which is on by default

* problems_mortar.jl: added new function diagnose_interface to calculate quantities from interface hopefully revealing bugs in calculation

* problems_mortar_3d.jl: added docstring for check_orientation! and removed flooding debug messages not helping to debug anything

* solvers.jl: Another way to solve Ax = b

* Refactored code to make implementation of Tri6 assemble! easier

* Patch test with linear Tet4 elements and quadratic Tet10 elements pass

When using quadratic elements, in polygon clipping algorithm element is divided to linear sub-elements as proposed in [Puso2008]. Interpolation of Lagrange multiplier space is done using quadratic shape functions.

References
----------

[Puso2008] Puso, Michael A., T. A. Laursen, and Jerome Solberg. "A segment-to-segment mortar contact method for quadratic elements and large deformations." Computer Methods in Applied Mechanics and Engineering 197.6 (2008): 555-566.

* increased coverage by adding diagnose_interface

* test using dual basis, failing for unknown reason

* Fixed dual basis construction for Mortar/Tet4

The coefficient matrix Ae for one particular slave element e is the result performing numerical integration on *all* integration cells associated with this element [Popp2013]. Ae cannot be calculated "cell-wise" like it was done before. Now patch test will pass also using `interface.properties.dual_basis = true` option. Partially integrated slave elements are supported as well.

References
----------

[Popp2013] Popp, Alexander, et al. "Improved robustness and consistency of 3D contact algorithms based on a dual mortar approach." Computer Methods in Applied Mechanics and Engineering 264 (2013): 67-80.

* Minor modifications to preprocess.jl

- removed two functions which are unimplemented (but maybe planned in future)
- added function create_node_set_from_element_set!, which can be used, like name suggests, to create a node set from nodes belonging to some set of elements.

* solvers.jl: now prints a list of overconstrained nodes which can be easily copy-pasted to problem.assembly.removed_dofs list to solver overconstrained situation manually

* Increase code coverage

Added a new test which tests dual basis 3d mortar + adjust option when using Tet4 in elasticity problem.

* Tet10 + Dual basis still failing, others are working

* mortar 3d low level tests

* linear surface element projection tests pass

* Introduced basis transform constant alpha

Tet10 + dual basis patch test still failing, but single element low level routine tests gives expected results with alpha=0.2

* added new integration rule FPG12 for triangular elements

* added drop_tolerance option to remove very small values from constraint matrices

* Introduced a basis transform matrix T

Constructing bi-orthogonal basis for quadratic surfaces is ill-conditioned. By doing a basis transform N' = N*T for slave side displacement vector it's possible to construct a bi-orthogonal basis in a same way than with linear elements. Setting alpha=0.2 ensures that quadratic basis functions are strictly positive in practical cases.

* fix 3d clipping test routine, accepts only 3d vertices

* dropped number of integration poitns from 12 to 7 in quadratic mortar surfaces intrestingly gives more accurate results, maybe something numerical error in FPG12 integration rule..?

* added two displacement patch tests + output writing for all cases

* %s/Int64/Int/g

* Changed test data location

* Fine tuning of logging levels

src/integrate.jl

@@ -148,23 +148,56 @@ function get_integration_points(element::TriangularElement, ::Type{Val{4}})
     return zip(weights, points)
 end
 
+""" 7 point integration rule for triangular elements.
+
+References
+----------
+Code Aster documentation, http://code-aster.org/doc/default/fr/man_r/r3/r3.01.01.pdf
+"""
 function get_integration_points(element::TriangularElement, ::Type{Val{5}})
-    weights = 0.5*[
-        0.22500000000000,
-        0.13239415278851,
-        0.13239415278851,
-        0.13239415278851,
-        0.12593918054483,
-        0.12593918054483,
-        0.12593918054483]
+    A = 0.470142064105115
+    B = 0.101286507323456
+    P1 = 0.066197076394253
+    P2 = 0.062969590272413
+    weights = [9/80, P1, P1, P1, P2, P2, P2]
     points = Vector{Float64}[
-        [0.33333333333333, 0.33333333333333], 
-        [0.47014206410511, 0.47014206410511], 
-        [0.47014206410511, 0.05971587178977],
-        [0.05971587178977, 0.47014206410511],
-        [0.10128650732346, 0.10128650732346],
-        [0.10128650732346, 0.79742698535309],
-        [0.79742698535309, 0.10128650732346]]
+        [1/3, 1/3], 
+        [A, A],
+        [1-2A, A],
+        [A, 1-2A],
+        [B, B],
+        [1-2B, B],
+        [B, 1-2B]]
+    return zip(weights, points)
+end
+
+""" 12 point integration fule for triangular elements.
+References
+----------
+Code Aster documentation, http://code-aster.org/doc/default/fr/man_r/r3/r3.01.01.pdf
+"""
+function get_integration_points{E<:TriangularElement}(element::Element{E}, ::Type{Val{:FPG12}})
+    A = 0.063089014491502
+    B = 0.249286745170910
+    C = 0.310352451033785
+    D = 0.053145049844816
+    P1 = 0.025422453185103
+    P2 = 0.058393137863189
+    P3 = 0.041425537809187
+    weights = [P1, P1, P1, P2, P2, P2, P3, P3, P3, P3, P3, P3]
+    points = Vector{Float64}[
+        [A, A],
+        [1-2A, A],
+        [A, 1-2A],
+        [B, B],
+        [1-2B, B],
+        [B, 1-2B],
+        [C, D],
+        [D, C],
+        [1-C-D, C],
+        [1-C,D, D],
+        [C, 1-C-D],
+        [D, 1-C-D]]
     return zip(weights, points)
 end