where
W =
1 4 2 4 . . . 2 4 1 4 16 8 16 . . . 8 16 4 2 8 4 8 . . . 4 8 2 4 16 8 16 . . . 8 16 4 ... ... ... ... . . . ... ... ... 2 8 4 8 . . . 4 8 2 4 16 8 16 . . . 8 16 4 1 4 2 4 . . . 2 4 1
(5.53)
Higher N lead to more precise results and smaller N to faster computations. We have compared the numerical results and the approximated results for different quadrilateral element geometries and N=10 provided good approximations for each case.
estimates of the basic solution variables (displacements in our case) and some flags defining the procedure type (see part5.5.3).
We have implemented three subroutines: for perfectly reflective, perfectly absorbent, and perfectly diffuse surfaces separately. In each case the solar pressure is characterized by three constants, the solar pressure modulated by the properties of the surfacepi(with i=1,2 or 3 depending on the surface type as in Equations 5.7, 5.8, 5.9), and the direction s = (s1, s2, s3) of the incoming light beam with s21+s22+s23 = 1. The subroutine properties are three positive or negative float numbers: pi,s1 ands2.
Effectively, the following parameters are used in the UEL subroutine: MCRD (number of d.o.f. per node)=3, NNODE (number of nodes per element)=4, and NDOFEL (total number of d.o.f. in an ele- ment)=12. The relation betweenrA and RHS andkAB and AMATRX are:
RHS(k1) =rA(ii) A= floor((k1−1)/3) + 1 ii= mod(k1−1,3) + 1
and
AM AT RX(k1, k2) =−kAB(ii, jj) A= floor((k1−1)/3) + 1
B= floor((k2−1)/3) + 1 ii= mod(k1−1,3) + 1 jj= mod(k2−1,3) + 1
The properties of the element are defined in the input file. We specified that the subroutine element has 4 nodes, unsymmetric stiffness matrix (use the option UNSYMM) and 3 properties to be defined. The degrees of freedom we use are the displacements which correspond to the components 1, 2, and 3. This corresponds to the following lines in the input file (a full input file can be found in AppendixDSectionD.3as an example).
*USER ELEMENT, NODES=4, UNSYMM, TYPE=U1, I PROPERTIES=0,PROPERTIES=3 1, 2, 3
The element properties are specified in the order (pi,s1,s2) as follows:
*UEL PROPERTY, ELSET=Set-2 0.01, 0.7071, 0.0
5.5.2 Tie Constraint
Now that we have developed an UEL subroutine we need to integrate it into Abaqus mechanical models.
We used tie constraints to add stiffness and nodal force from solar pressure element to an existing Abaqus
model. The idea is to create a new 3D surface part that matches the surface of the object on which the load is applied both in geometry and mesh. As we have implemented the subroutine for quadrilateral elements the surface of the object needs to be meshed accordingly. One can use for example hexahedral continuum shell elements for thin objects or hexahedral elements for 3D objects. This is illustrated Figure5.4afor thin continuum shell elements and Figure5.4b for solid elements. The load elements are in red and mechanical elements from Abaqus libraries in green.
Load Elements
Mechanical Elements Tie Constraint
at Every Node
(a) Thin continuum shell mechanical elements.
Load Elements
Mechanical Elements Tie Constraint
at Every Node
(b) Solid mechanical elements.
Figure 5.4: The red elements are the load elements whose geometry and mesh match the object (in green) surface. There is effectively no separation between red and green surfaces. The thin dotted lines represent the tie constraint at every node.
Since the load elements are surface elements, a convention on the positive normal direction needs to be introduced. Abaqus by default uses the convention in Figure 5.5. Since the normal Equation 5.20might not be in the proper direction (relative to the object these surface elements are attached to), the direction smust be chosen such that the overall force on the object is in the modeled direction.
Z
Y X
n
1 2
4 3
Figure 5.5: Abaqus surface elements convention.
The load and mechanical parts are then connected using a surface based tie constraint: the mesh tie constraint (Simulia, 2013a). Each node on the slave (load) surface is constrained to have the same motion
as the point on the master (mechanical) surface to which it is closest (spot on in our case). We use a nodal region and only constrain the translational degrees of freedom. We implement all boundary conditions and loads (except solar pressure) on the mechanical (master) surface only.
5.5.3 Abaqus Analysis
We want to use this method to estimate equilibria and run frequency-based stability analyses. The subroutine was thus implemented and tested for static and complex frequency analyses. In particular, the amplitude of the pressure is implemented differently for each case and we use flags to differentiate the type of analysis at each step.
5.5.3.1 Static Analysis
For the static analysis and in particular in the nonlinear case the amplitude of the loadpiis linearly increased within the step in order to improve the convergence of the Newton-Raphson algorithm. To this end the target amplitude is multiplied by the current time of the step and the step total time is 1. The flag to be used to identify this analysis is LFLAGS(1).
With this convention, by default solar radiation pressure is incrementally increased at every static step.
The static step at which solar pressure is not required or is maintained constant should be identified in the subroutine (using KSTEP identifier) andpi should respectively be set to 0 and 1. Additionally, by default Abaqus makes the stiffness matrix of the system symmetric. We need to specify an unsymmetric stiffness matrix during the static step using the option unsymm=YES.
5.5.3.2 Complex Frequency Analysis
As the load stiffness can be unsymmetric, the frequency and mode shapes of the structure can be complex.
Abaqus can evaluate complex frequency and this analysis type is compatible with UEL subroutines (but does not account for damping from the subroutine, which is not a restriction for our application). In this case, the amplitude of the load pi is directly specified and not incrementally increased. The flags for frequency and complex frequency analysis are respectively 41 and 47.
The procedure to obtain the complex frequencies is a subspace projection method. First the eigenmodes of the symmetric stiffness matrix are extracted in a frequency analysis. Then the complex eigenfrequency extraction is performed. In order to obtain accurate results, a large enough set of eigenmodes needs to be extracted prior to the complex frequency step and a convergence analysis needs to be carried out.
We have tested this implementation in Abaqus/Standard in the case of a perfectly reflective surface as shown next.