- problem can be now represented using potential energy or residual

force vector, autodiff takes care of linearization

- elasticity equations are now solved using e.g. principle of minimum
  potential energy. syntax is quite good, see notebook.

- updated how to interpolate fields, by introducing function spaces.
  syntax is now good. still have to figure out how to do time derivatives

- etc. etc. tutorial is broken at the moment, i took of get_lhs and
  get_rhs because they didn't really work.

test/test_interpolation.jl

@@ -0,0 +1,50 @@
+# This file is a part of JuliaFEM.
+# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
+
+using JuliaFEM: get_basis, grad, FieldSet, Field, Quad4
+using FactCheck
+
+element = Quad4([1, 2, 3, 4])
+
+geometry_field = Field(0.0, Vector[])  # Create empty field at time t=0.0
+push!(geometry_field, [ 0.0, 0.0])  # push some values for field
+push!(geometry_field, [ 1.0, 0.0])
+push!(geometry_field, [ 1.0, 1.0])
+push!(geometry_field, [ 0.0, 1.0])
+geometry_fieldset = FieldSet("geometry")  # create fieldset "geometry"
+push!(geometry_fieldset, geometry_field)  # add field to fieldset
+push!(element, geometry_fieldset) # add fieldset to element
+
+temperature_fieldset = FieldSet("temperature")
+push!(temperature_fieldset, Field(0.0, [0.0, 0.0, 0.0, 0.0]))
+push!(temperature_fieldset, Field(1.0, [1.0, 2.0, 3.0, 4.0]))
+push!(element, temperature_fieldset)
+
+displacement_fieldset = FieldSet("displacement")
+push!(displacement_fieldset, Field(0.0, Vector[[0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0]]))
+push!(displacement_fieldset, Field(1.0, Vector[[0.0, 0.0], [0.0, 0.0], [0.25, 0.0], [0.0, 0.0]]))
+push!(element, displacement_fieldset)
+
+facts("basic continuum interpolations") do
+    # from my old home works
+    basis = get_basis(element)
+    dbasis = grad(basis)
+    @fact basis("geometry", [0.0, 0.0], 1.0) + basis("displacement", [0.0, 0.0], 1.0) --> [9/16, 1/2]
+    gradu = dbasis("displacement", [0.0, 0.0], 1.0)
+    epsilon = 1/2*(gradu + gradu')
+    rotation = 1/2*(gradu - gradu')
+    X = basis("geometry", [0.0, 0.0], 1.0)
+    k = 0.25
+    epsilon_wanted = [X[2]*k 1/2*X[1]*k; 1/2*X[1]*k 0]
+    rotation_wanted = [0 k/2*X[1]; -k/2*X[1] 0]
+    @fact epsilon --> roughly(epsilon_wanted)
+    @fact rotation --> roughly(rotation_wanted)
+    F = I + gradu
+    @fact F --> [X[2]*k+1 X[1]*k; 0 1]
+    C = F'*F
+    @fact C --> [(X[2]*k+1)^2  (X[2]*k+1)*X[1]*k; (X[2]*k+1)*X[1]*k  X[1]^2*k^2+1]
+    E = 1/2*(F'*F - I)
+    @fact E --> [1/2*(X[2]*k + 1)^2-1/2  1/2*(X[2]*k+1)*X[1]*k; 1/2*(X[2]*k + 1)*X[1]*k  1/2*X[1]^2*k^2]
+    U = 1/sqrt(trace(C) + 2*sqrt(det(C)))*(C + sqrt(det(C))*I)
+    #@fact U --> roughly([1.24235 0.13804; 0.13804 1.02149])
+end