Users:General FEM Analysis/Analyses Reference/Cutting Pattern

From Carat++ Public Wiki
< Users:General FEM Analysis | Analyses Reference(Difference between revisions)
Jump to: navigation, search
(Created page with "Category: Users:General FEM Analysis == General Description == The static nonlinear analysis is applied to analyze displacements, strains or stresses of nonlinear structure...")
 
(→Parameter Description)
 
(30 intermediate revisions by 2 users not shown)
Line 3: Line 3:
 
== General Description ==
 
== General Description ==
  
−
The static nonlinear analysis is applied to analyze displacements, strains or stresses of nonlinear structures. There exists several sources of nonlinearities like nonlinear kinematics, nonlinear material behavior or contact scenarios.
+
Cutting pattern generation determines a plane and (optimally) stress-free geometry which can be reassembled to build a membrane with a certain prestress and shape (as it was determined in Formfinding).<ref name="Dieringer"> Dieringer, F.: Numerical Methods for the Design and Analysis of Tensile Structures, Dissertation Lehrstuhl für Statik, 2014 </ref>
  
−
The static nonlinear analysis formulates structural equilibrium on the actual configuration under consideration of the actual system response.  
+
Prerequisites: [[Users:Form_Finding|Formfinding]] and [[Users:General_FEM_Analysis/Analyses_Reference/Geodesic_Lines|Geodesic Line Search]]
 +
 
 +
'''Note''' that an instruction is included in the "Membran-Workshop" documents.
  
 
=== Structure of Equation System ===
 
=== Structure of Equation System ===
  
−
A nonlinear problem is formulated by the equation '''r''' = '''f'''_int('''u''') - λ * '''f'''_ext('''u''') where '''r''' specifies the residual vector and '''f'''_int and '''f'''_ext define the internal and external forces respectively. In general, the internal forces as well as the external forces depend on the actual displacement field '''u'''. Thus, the equation is nonlinear with respect to the a priori unknown equilibrium displacements. The parameter λ specifies a scaling parameter for the external load. Usually this parameter is controlled by a load curve which specifies λ as a function of the pseudo time t. The slope of this load curve must be small enough such that convergence is possible in each step. Whenever a nonlinear analysis has convergence problems it is appropriate to reduce the slope of this load curve and calculate more time steps.
+
A nonlinear problem is formulated by the equation '''r''' = '''f'''_int('''u''') where '''r''' specifies the residual vector and '''f'''_int defines the internal forces respectively. In general, the internal forces depend on the actual displacement field '''u'''. Thus, the equation is nonlinear with respect to the a priori unknown equilibrium displacements.   
−
 
+
−
At the equilibrium point the internal forces are equal to the external forces and the residual vector is equal to zero. The above specified nonlinear problem is linearized for the actual displacement state and solved by a Newton-Raphson scheme where the residual vector is used to compute incremental displacements by '''K'''_t '''u'''_inc = '''r'''. Here the tangential stiffness matrix is specified by '''K'''_t whereas '''u'''_inc denotes the incremental displacements.
+
−
 
+
−
=== Path Following Methods ===
+
−
The iterative solution of the nonlinear problem requires path following methods. The most simple path following method is a pure '''Load Control'''. Here the parameter ''lambda'' is increased according to the specified load curve. This allows nonlinear analysis for relatively robust nonlinear problems. But as soon as the structure shows instabilities the load control method usually does not not converge anymore. In this case the '''Arc Length''' method can be applied to compute the equilibrium path. This method controls the so called arc length which is defined as a combination from displacement increment and load increment. In general it is also possible to directly control a specified displacement. But actually this path control method is not available in Carat++. More information about path following methods can be found in the thesis of Reiner Reitinger<ref name="Rei94">Reitinger, R.: Stabilität und Optimierung imperfektionsempfindlicher Tragwerke, Dissertation, Bericht Nr. 17, Institut für Baustatik, Universität Stuttgart, 1994</ref> and Amphon Jarusjarungkiat.
+
−
 
+
−
=== Step Length Control ===
+
−
Actually there exist two methods for step length control in Carat++. The most simple method is the fixed step length which computes the whole load displacement path with a constant step length. This procedure is robust but inefficient because a large number of load steps has to be computed. Usually it is more efficient to apply adaptive step length methods. A very robust method was presented by Crisfield and Ramm, c.f. <ref name="Rei94" />. This method specifies the actual step length for the time step ''i'' by the operation ''StepLength_i'' = ''alpha'' * ''StepLength_i-1''. The parameter ''alpha'' is computed by ''(J_d / (J_i-1 * (Nmb_Restarts+1)))^p'' with: ''J_d'' = desired number of equilibrium iterations, ''J_i-1'' = equilibrium iterations of last time step and ''p'' equals the exponent.
+
−
 
+
−
<pre>
+
−
Example: Actual iteration step:                                  i
+
−
          Number of desired equilibrium iterations:                J_d = 8.
+
−
          Number of equilibrium iterations in the last time step:  J_i-1 = 12.
+
−
          Number of restarts:                                      Nmb_Restarts = 0.
+
−
          Exponent:                                                p=1.0.
+
−
          Step length of last step:                                StepLength_i-1 = 0.2.
+
−
 
+
−
          alpha = ( 8 / (12*(0+1)))^(1.0) = (8 / 12)^(1.0) = 2/3.
+
−
          StepLength_i = 2/3 * 0.2 = 0.1333.
+
−
 
+
−
</pre>
+
−
 
+
−
Thus, if the number of equilibrium iterations is larger than a desired value the step length is decreased for the next time step. If the number of equilibrium iterations is smaller than a desired value the step length is increased for the next time step. The exponent ''p'' can be used to manipulate the update speed. It should be used with care.
+
−
 
+
−
=== Simultaneous Eigenvalue Analysis ===
+
−
Reaching critical points of the load displacement path (limit points, bifurcation points, ...) requires special investigations. A simultaneous eigenvalue analysis solving the problem ('''K'''_t - ''eigenvalue'' * '''I''') '''phi''' = '''0''' allows for approximation of the load factor at the next critical point ''lambda''_k. Whenever a critical point is reached the above equations yields to a zero eigenvalue. Extrapolating the current load factor according to the eigenvalue allows for good approximations of the critical load.
+
  
−
=== Imperfect Designs ===
+
At the equilibrium point the residual vector is equal to zero. The above specified nonlinear problem is linearized for the actual displacement state and solved e.g. by a Newton-Raphson scheme where the residual vector is used to compute incremental displacements by '''K'''_t '''u'''_inc = '''r'''.
−
Many structures subjected to large compression loading are sensitive to geometrical imperfections. Unfortunately the shape of the imperfection is a priori unknown but mostly some information about tolerances exist. In the nonlinear analysis the imperfection shapes can be computed as sum of several  normalized eigenmodes. The final imperfection mode is then scaled by the specified tolerance. This operation can be expressed by d'''x'''_imp = tol * (sum_i '''phi'''_i). After computation of the imperfection mode, the geometry of the structure is modified and the nonlinear analysis starts with the imperfect geometry.
+
  
 
== Input Parameters ==
 
== Input Parameters ==
Line 52: Line 27:
 
|-
 
|-
 
!PC-ANALYSIS
 
!PC-ANALYSIS
−
|''int'' : STA_GEO_NONLIN
+
|''int'' : CUTTING_PATTERN
−
|Keyword of nonlinear analysis with analysis ID
+
|Keyword of analysis with analysis ID
−
|-
+
−
!PATHCONTROL
+
−
|FORCE or ARCLENGTH or DISPLACEMENT
+
−
|Definition of path control method. (DISPLACEMENT is actually not available.)
+
 
|-
 
|-
 
!SOLVER
 
!SOLVER
Line 74: Line 45:
 
|EL-DOMAIN ''int''
 
|EL-DOMAIN ''int''
 
|Linking to the domain the analysis should work on
 
|Linking to the domain the analysis should work on
−
|-
 
−
!NUM_STEP
 
−
|''int''
 
−
|Number of time steps that have to be calculated
 
 
|-
 
|-
 
!MAX_ITER_EQUILIBRIUM
 
!MAX_ITER_EQUILIBRIUM
Line 87: Line 54:
 
|Equilibrium accuracy that has to be reached for convergence. The convergence is checked with the L2 norm of the incremental displacements.
 
|Equilibrium accuracy that has to be reached for convergence. The convergence is checked with the L2 norm of the incremental displacements.
 
|-
 
|-
−
!CURVE
+
!PATTERN
−
|LD-CURVE ''int''
+
|PART ''int''
−
|Linking to the load curve.
+
|The parts which should be included in the analysis, separation with comma.
 
|-
 
|-
−
|colspan="3" style="background:#efefef;"| Optional Parameters
+
!BC_CONFIGURATION
 +
|TYPE ''int''
 +
|The type of boundary condition configuration, for each part one type must be given. 1... first and last node, 2... last two nodes in the list, 3... first two nodes in the list, 4... largest distance between nodes.
 
|-
 
|-
−
!TRACED_NODE
+
!RELAXATION_METHOD
 +
|STATIC or NONE or GALERKIN or COMBINED
 +
|Definition of relaxation method.
 +
|-
 +
!FLATTENING_AREA
 +
|0 or 1 or 2
 +
|0... prestress area. 1... mean surface normal. 2... cylinder with RADIUS and RO_X,RO_Y,RO_Z.
 +
|-
 +
!PATTERNING_METHOD
 +
|GALERKIN or LS_NR or LS_CG or NONE
 +
|Definition of the patterning method. GALERKIN... principle of virtual work. LS_NR and LS_CG... optimization problem solved with Newton-Raphson or Conjugate Gradient approach.
 +
|-
 +
|colspan="3" style="background:#efefef;"| Optional parameters for the consideration of seam lines
 +
|-
 +
!EQUAL_SEAM_LINES
 +
|TRUE or FALSE
 +
|Includes the seam lengths into the cutting pattern analysis and minimizes the length difference of common edges.
 +
|-
 +
!PENALTY_FAC
 
|''int''
 
|''int''
−
|Node ID of traced node. Used for screen and log file output.
+
|Definition of a penalty factor for the seam line length optimization.
 
|-
 
|-
−
!TRACED_NODAL_DOF
+
!SEAM_LINES
−
|''dof type''
+
|TRUE or FALSE
−
|DOF type of node TRACED_NODE that has to be traced (DISP_X, DISP_Y, DISP_Z, ...).
+
|Includes the increased stiffness along the seam lines into the analysis.
 
|-
 
|-
−
!STEP_LENGTH_CONTROL
+
!SEAM_AREA
−
|FIXED or CRISFIELD_RAMM
+
|''float''
−
|Type of step length control algorithm (for ARCLENGTH method). FIXED specifies a constant step size where the desired step size is defined by the parameter STEP_LENGTH_CONTROL_REALS. CRISFIELD_RAMM specifies an adaptive method according to Crisfield and Ramm, c.f. <ref name="Rei94" />. In this case the parameter  STEP_LENGTH_CONTROL_INTS = ''int'' specifies the number of restarts that are allowed. The other necessary parameters are specified by STEP_LENGTH_CONTROL_REALS. See the example below.
+
|Definition of the seam area for an analysis which includes the seam lines.
 
|-
 
|-
−
!STEP_LENGTH_CONTROL_REALS
+
!SEAM_MATERIAL
−
|''float'', ''float'', ...
+
|''int''
−
|respective number of floats for step length control algorithm
+
|Definition of the seam line material for an analysis which includes the seam lines.
 
|-
 
|-
−
!STEP_LENGTH_CONTROL_INTS
 
−
|''int'', ''int'', ...
 
−
|respective number of integers for step length control algorithm
 
 
|-
 
|-
−
!SIMULTANEOUS_EIGENVALUE_ANALYSIS
+
|colspan="3" style="background:#efefef;"| Optional parameters for a cylinder projection before flattening the stripes
−
|1 or 0
+
−
|flag to specify if a simultaneous eigenvalue analysis should be performed, 0 - no eigenvalue analysis, 1 - perform eigenvalue analysis. This flag requires definition of parameter EIGEN_SOLVER.
+
 
|-
 
|-
−
!EIGEN_SOLVER
+
!RADIUS
−
|PC-SOLVER ''int''
+
|''float''
−
|Linking to an eigenvalue solver
+
|If a cylinder is defined for the projection area (FLATTENING_AREA=2) Definition of the cylinder radius.
 +
|-
 +
!RO_X; RO_Y; RO_Z
 +
|''float''
 +
|Definition of the root point for the cylinder as a projection area.
 
|-
 
|-
−
!IMPERFECTION_MODES
 
−
|''int'', ''int'', ''int'', ...
 
−
|specifies which eigen modes should be summarized to the imperfection mode. Computation of imperfect design requires definition of parameter EIGEN_SOLVER.
 
 
|-
 
|-
−
!IMPERFECTION_SIZE
+
|colspan="3" style="background:#efefef;"| Optional parameters for a possible restart analysis after the cutting-pattern generation
−
|''float''
+
|-
−
|specifies the size of the imperfection mode.
+
!MOUNTING_ANALYSIS
 +
|''int''
 +
|Starts the mounting analysis after the cutting pattern generation.
 +
|-
 +
!MOUNTING_ANALYSIS_TYPE
 +
|FULL or RESTART or INCREMENTAL
 +
|Definition of the mounting analysis type.
 
|}
 
|}
  
Line 133: Line 122:
  
 
<pre>
 
<pre>
−
PC-ANALYSIS 1: STA_GEO_NONLIN
+
PC-ANALYSIS 1: CUTTING_PATTERN
−
   PATHCONTROL = ARCLENGTH ! or DISPLACEMENT or ARCLENGTH
+
   DOMAIN = EL-DOMAIN 1
 +
  MAX_ITER_EQUILIBRIUM = 20
 +
  EQUILIBRIUM_ACCURACY = 1e-06
 +
  COMPCASE = LD-COM 1
 +
  OUTPUT  = PC-OUT 1
 +
  PATTERN = PART 1,3
 +
  BC_CONFIGURATION = TYPE 4,4
 
   SOLVER = PC-SOLVER 1
 
   SOLVER = PC-SOLVER 1
−
   OUTPUT = PC-OUT 1
+
   RELAXATION_METHOD=STATIC    !NONE, STATIC, GALERKIN, COMBINED
−
   COMPCASE = LD-COM 1
+
   FLATTENING_AREA=1           !0 ...Prestress area, 1 ...Mean surface normal
−
  DOMAIN = EL-DOMAIN 1
+
   PATTERNING_METHOD=GALERKIN  !LS_NR, GALERKIN, LS_CG, NONE
−
  NUM_STEP = 30
+
−
  MAX_ITER_EQUILIBRIUM = 100
+
−
  EQUILIBRIUM_ACCURACY = 1e-10
+
−
  CURVE=LD-CURVE 1
+
−
  TRACED_NODE=2
+
−
  TRACED_NODAL_DOF=DISP_Y
+
−
  ! Example: fixed step length of 0.05
+
−
    STEP_LENGTH_CONTROL = FIXED
+
−
    STEP_LENGTH_CONTROL_REALS = 0.05        ! constant step length of 0.05
+
−
  ! Example adaptive step length according to Crisfield and Ramm
+
−
    STEP_LENGTH_CONTROL = CRISFIELD_RAMM
+
−
    STEP_LENGTH_CONTROL_REALS = 0.1, 1.0    ! initial step length = 0.1, exponent p = 1.0
+
−
    STEP_LENGTH_CONTROL_INTS = 5            ! 5 equilibrium iterations per step are desired
+
−
   ! Example: no simultaneous eigenvalue analysis
+
−
    SIMULTANEOUS_EIGENVALUE_ANALYSIS = 0
+
−
  ! Example: simultaneous eigenvalue analysis
+
−
    SIMULTANEOUS_EIGENVALUE_ANALYSIS = 1
+
−
    EIGEN_SOLVER = PC-SOLVER 2
+
−
  ! Example: use imperfect design with modes 1, 2, 4, and 6
+
−
    IMPERFECTION_MODES = 1, 2, 4, 6
+
−
    IMPERFECTION_SIZE = 0.015                ! maximum size of imperfection mode equals 0.015
+
−
    EIGEN_SOLVER = PC-SOLVER 2
+
 
</pre>
 
</pre>
  
 
== Example ==
 
== Example ==
  
−
The following simple example shows a geometrically nonlinear analysis of a 2bar truss system using the arclength method. It uses a fixed step size for computation.
+
The following simple example shows the cutting pattern analysis of two parts from a four-point sail. It uses a static relaxation method, the flattening area is the mean surface normal and a Galerkin approach is used for the patterning method.
−
The respective input file can be found in the SVN repository under
+
−
<pre>
+
−
carat20/examples/benchmark_examples/analyses/stanln_2bartruss_arclength_l/2bartruss_arclength_fixed.txt
+
−
</pre>
+
−
The problem computes a snap through problem. The [[Users:General FEM Analysis/BCs Reference|boundary conditions]] are visualized by the figure below.
+
 
{|
 
{|
−
|[[File:2bar_truss_org.png‎|frame|up|Simple 2bar truss structure discretized with [[Users:General FEM Analysis/Elements Reference/Truss1|truss elements]]]]  
+
|[[File:Benchmark_cutpat_4point.png]]
 
|}
 
|}
−
The load f is increased by the arclength method such that the displacements depicted in the figure below occur.
+
The cutting pattern of the two stripes is shown in the picture below.
 
{|
 
{|
−
|[[File:2bar_truss_defo.gif|frame|up|Truss snap through]]
+
|[[File:Benchmark_cutpat_4point_relaxed.png]]  
−
|}
+
−
The figure below shows the load displacement path of the y-displacement of the center node. The well known snap through behavior is clearly visible. For a load factor of approx. 0.14 the structure shows instability. In the following the arclength control reduces the load factor until -0.14 to compute equilibrium states. Obviously these states can be only be computed by arclength or displacement control. After a complete snap through the load can be increased further. Now the complete structure works in tension which does not yield to further instability points.
+
−
{|
+
−
|[[File:2bar_truss_load_disp_curve.png|frame|up|Load displacement path of 2bar truss]]  
+
 
|}
 
|}
 +
 +
=== Benchmark examples ===
 +
* four point sail with coarse discretization: ..\examples\benchmark_examples\analyses\cutPat_Membrane1\cbm_CutPat.dat
 +
* four point sail with coarse discretization and automatic mounting analysis after restart: ..\examples\benchmark_examples\analyses\cutPatWithMounting_Membrane1\cbm_CutPatWithMounting.dat
 +
* four point sail with linear elastic material and membrane 1 elements (not included in regular benchmarks due to amount of time for running, but working!) ..\examples\benchmark_examples\analyses\cutting_patterning_membrane1_I\x_cbm_4_point_patterning.txt
  
 
== References ==
 
== References ==
  
 
<references/>
 
<references/>

Latest revision as of 08:16, 27 April 2020


Contents

General Description

Cutting pattern generation determines a plane and (optimally) stress-free geometry which can be reassembled to build a membrane with a certain prestress and shape (as it was determined in Formfinding).[1]

Prerequisites: Formfinding and Geodesic Line Search

Note that an instruction is included in the "Membran-Workshop" documents.

Structure of Equation System

A nonlinear problem is formulated by the equation r = f_int(u) where r specifies the residual vector and f_int defines the internal forces respectively. In general, the internal forces depend on the actual displacement field u. Thus, the equation is nonlinear with respect to the a priori unknown equilibrium displacements.

At the equilibrium point the residual vector is equal to zero. The above specified nonlinear problem is linearized for the actual displacement state and solved e.g. by a Newton-Raphson scheme where the residual vector is used to compute incremental displacements by K_t u_inc = r.

Input Parameters

Parameter Description

Compulsory Parameters
Parameter Values, Default(*) Description
PC-ANALYSIS int : CUTTING_PATTERN Keyword of analysis with analysis ID
SOLVER PC-SOLVER int Linking to a linear solver (direct or iterative)
OUTPUT PC-OUT int Linking to output objects (specifies the type of output format, e.g. GiD)
COMPCASE LD-COM int Linking to computation case object which specify the boundary conditions (loading and supports). Only a single computation case is allowed.
DOMAIN EL-DOMAIN int Linking to the domain the analysis should work on
MAX_ITER_EQUILIBRIUM int Maximum number of equilibrium iterations that are allowed.
EQUILIBRIUM_ACCURACY float Equilibrium accuracy that has to be reached for convergence. The convergence is checked with the L2 norm of the incremental displacements.
PATTERN PART int The parts which should be included in the analysis, separation with comma.
BC_CONFIGURATION TYPE int The type of boundary condition configuration, for each part one type must be given. 1... first and last node, 2... last two nodes in the list, 3... first two nodes in the list, 4... largest distance between nodes.
RELAXATION_METHOD STATIC or NONE or GALERKIN or COMBINED Definition of relaxation method.
FLATTENING_AREA 0 or 1 or 2 0... prestress area. 1... mean surface normal. 2... cylinder with RADIUS and RO_X,RO_Y,RO_Z.
PATTERNING_METHOD GALERKIN or LS_NR or LS_CG or NONE Definition of the patterning method. GALERKIN... principle of virtual work. LS_NR and LS_CG... optimization problem solved with Newton-Raphson or Conjugate Gradient approach.
Optional parameters for the consideration of seam lines
EQUAL_SEAM_LINES TRUE or FALSE Includes the seam lengths into the cutting pattern analysis and minimizes the length difference of common edges.
PENALTY_FAC int Definition of a penalty factor for the seam line length optimization.
SEAM_LINES TRUE or FALSE Includes the increased stiffness along the seam lines into the analysis.
SEAM_AREA float Definition of the seam area for an analysis which includes the seam lines.
SEAM_MATERIAL int Definition of the seam line material for an analysis which includes the seam lines.
Optional parameters for a cylinder projection before flattening the stripes
RADIUS float If a cylinder is defined for the projection area (FLATTENING_AREA=2) Definition of the cylinder radius.
RO_X; RO_Y; RO_Z float Definition of the root point for the cylinder as a projection area.
Optional parameters for a possible restart analysis after the cutting-pattern generation
MOUNTING_ANALYSIS int Starts the mounting analysis after the cutting pattern generation.
MOUNTING_ANALYSIS_TYPE FULL or RESTART or INCREMENTAL Definition of the mounting analysis type.

Example of a Complete Input Block 

PC-ANALYSIS 1: CUTTING_PATTERN
  DOMAIN = EL-DOMAIN 1
  MAX_ITER_EQUILIBRIUM = 20
  EQUILIBRIUM_ACCURACY = 1e-06
  COMPCASE = LD-COM 1
  OUTPUT   = PC-OUT 1
  PATTERN = PART 1,3
  BC_CONFIGURATION = TYPE 4,4
  SOLVER = PC-SOLVER 1
  RELAXATION_METHOD=STATIC    !NONE, STATIC, GALERKIN, COMBINED
  FLATTENING_AREA=1           !0 ...Prestress area, 1  ...Mean surface normal				
  PATTERNING_METHOD=GALERKIN  !LS_NR, GALERKIN, LS_CG, NONE

Example

The following simple example shows the cutting pattern analysis of two parts from a four-point sail. It uses a static relaxation method, the flattening area is the mean surface normal and a Galerkin approach is used for the patterning method.

Benchmark cutpat 4point.png

The cutting pattern of the two stripes is shown in the picture below.

Benchmark cutpat 4point relaxed.png

Benchmark examples

  • four point sail with coarse discretization: ..\examples\benchmark_examples\analyses\cutPat_Membrane1\cbm_CutPat.dat
  • four point sail with coarse discretization and automatic mounting analysis after restart: ..\examples\benchmark_examples\analyses\cutPatWithMounting_Membrane1\cbm_CutPatWithMounting.dat
  • four point sail with linear elastic material and membrane 1 elements (not included in regular benchmarks due to amount of time for running, but working!) ..\examples\benchmark_examples\analyses\cutting_patterning_membrane1_I\x_cbm_4_point_patterning.txt

References

  1. ↑ Dieringer, F.: Numerical Methods for the Design and Analysis of Tensile Structures, Dissertation Lehrstuhl für Statik, 2014




Whos here now:   Members 0   Guests 0   Bots & Crawlers 1
 
Retrieved from "http://carat.st.bv.tum.de/caratuserswiki/index.php/Users:General_FEM_Analysis/Analyses_Reference/Cutting_Pattern"
Personal tools
Content for Users
Content for Developers