Merge pull request #28212 from grmnptr/dim-reduction-28200

Add dimensionality reduction example with multioutput GPs for 2D meltpools simulations

modules/combined/doc/content/modules/combined/examples/stm_laserwelding_dimred.md

@@ -0,0 +1,172 @@
+# Using Stochastic Tools for the full-field reconstruction of multiphysics problems
+
+The purpose of this example is to present a multiphysics model order reduction case using
+the [Stochastic Tools Module](modules/stochastic_tools/index.md). The problem of interest is a
+melt pool model using the [Navier-Stokes](modules/navier_stokes/index.md) module.
+The problem involves multiple uncertain process parameters and are interested in predicting
+full field quantities for unseen process parameter combinations using non-intrusive surrogate
+models in combination with Proper Orthogonal Decomposition (POD).
+
+## Problem Description
+
+The problem of interest involves a transient 2D melt pool simulation for a 316L stainless steel
+with a moving Gaussian laser beam. The laser source melts the top layer of the metal and
+based on the equlibrium of the forces (surface tension, vapor recoil pressure) the surface of the
+melt pool can deform considerably. The changing geometry is taken into account using an
+Arbitrary Legrangian Eulerian (ALE) approach. The setup of the problem is depicted in [fig:geom].
+
+!media combined/laser_weld/geometry.png caption=Problem setup for the laser melt pool simulation id=fig:geom style=width:100%
+
+The temperature-dependent thermal properties of the problem have been taken from the
+references listed in [tab:mat_prop].
+
+!table caption=Material properties for 316L Stailess steel id=tab:mat_prop
+| Property | Source|
+| :- | :- |
+| Dynamic viscosity | [!cite](kim1975thermophysical)$^\dagger$ |
+| Density | [!cite](kim1975thermophysical) |
+| Specific heat | [!cite](kim1975thermophysical) |
+| Thermal conductivity | [!cite](pichler2022surface) |
+| Surface tension | [!cite](pichler2022surface) |
+| Vapor recoil pressure | [!cite](chen2021numerical) |
+
+$\dagger$ In the treatment of dynamic viscosity, following the recommendations in [!cite](noble2007use),
+we artificially cap the minimum value to stabilize the fluid flow solver.
+
+The input file used for the simulations is presented below:
+
+!listing examples/stochastic/laser_welding_dimred/2d.i caption=Laser meltpool model input file id=list:meltpool-ref
+
+We see that it utilizes the `!include` syntax to read the process parameters togheter with the mesh and
+kernel and boundary condition objects from other input files. This input file design reduces
+duplication when it comes to the testing of the reduced-order models in subsequent sections.
+
+!listing examples/stochastic/laser_welding_dimred/parameters.i caption=Input file for specifying process parameters id=list:parameters-ref
+
+!listing examples/stochastic/laser_welding_dimred/mesh.i caption=Input file for specifying the mesh id=list:mesh-ref
+
+!listing examples/stochastic/laser_welding_dimred/physics_objects.i caption=Input file adding the objects that define the physics id=list:physics-ref
+
+### Process parameters
+
+A total of two process parameters have been identified to be changable: the effective laser power and
+the effective laser radius. They can change in the intervals shown in [tab:ppars] assuming a uniform
+distribution.
+
+!table caption=The distributions used for the process parameters id=tab:ppars
+| Property | Min. | Max. |
+| :- | - | - |
+| Effective laser power  | 60 W | 74 W |
+| Effective laser radius | 125 $\mu m$ | 155 $\mu m$ |
+
+### Quantities of Interest
+
+The quantity of interest in our case is the whole temperature field at the end of the simulation (0.4~ms).
+To be able to reconstruct the whole field we utilize Proper Orthogonal Decomposition (POD) in conjunction
+ with a Multi-output Gaussian Process
+(MOGP).
+
+## Results
+
+## Reference results
+
+Two reference results are presented here:
+1. Results for a simulation with low effective laser power ($60~W$) and wide effective laser radius ($155~\mu m$)
+2. Results for a simulation with high effective laser power ($74~W$) and narrow effective laser radius ($125~\mu m$)
+
+The reference results are filtered to show the melt pool shape. We see that for the first case, the
+steel melts, but no displacement is visible due to little to no evaporation. On the other hand, the
+increase in laser power results in an increased evaporation resulting in significant surface deformation.
+
+!media combined/laser_weld/60.png caption=Simulation results at the final time step with laser power ($60~W$) and wide effective laser radius ($155~\mu m$) id=fig:result-60 style=width:100%
+
+!media combined/laser_weld/74.png caption=Simulation results at the final time step with laser power ($74~W$) and narrow effective laser radius ($125~\mu m$) id=fig:result-74 style=width:100%
+
+## Training the reduced-order model
+
+The reduced-order model was trained by first running the reference model with 45 different
+process parameter combinations in a [Monte Carlo sampler](MonteCarloSampler.md). The input file used for training
+is presented below.
+
+!listing examples/stochastic/laser_welding_dimred/train.i id=list:training
+
+The parts that require special attention are the snapshot gathering
+parts in the transfer and reporter blocks.
+
+!listing examples/stochastic/laser_welding_dimred/train.i caption=Transfers and Reporters needed for snapshot collection block=Transfers Reporters VariableMappings remove=Reporters/matrix Reporters/svd id=list:training-trans-rep
+
+With the snapshots all available in one place, we can extract the POD modes
+necessary for the dimensionality reduction. This is done using the
+[PODMapping.md] reporter which not only creates the decomposition but also maps the
+created snapshots onto a low-dimensional latent space.
+
+!plot scatter
+  id=results caption=Scree plot of the first 8 singular values extracted from the training data.
+  filename=examples/stochastic/laser_welding_dimred/gold/sv_to_print.csv
+  data=[{'x':'i', 'y':'sv', 'name':'Singular values'}]
+  layout={'xaxis':{'type':'linear', 'title':'Mode #'},
+          'yaxis':{'type':'log','title':'Singular Value (-)'}}
+
+ In this specific example, the
+dimensionality of the latent space was selected to be 8.
+Once the coordinates of the snapshots are available in the latent space, we can fit a
+[MOGP](GaussianProcessTrainer.md) on them. This is done in the `Trainers` block with the corresponding covariance functions defined in the `Covariance` block.
+
+!listing examples/stochastic/laser_welding_dimred/train.i block=Trainers Covariance id=list:training-gp caption=Definition of the Gaussian Process surrogate used for fitting the coordinates in the latent space.
+
+One can then save both the POD mapping and MOGP into restratable structures for
+later usage.
+
+!listing examples/stochastic/laser_welding_dimred/train.i block=Outputs
+
+## Evaluating the error of the ROM
+
+We evaluate the ROM by comparing reconstructed temperature fields at unseen process
+parameter combinations to full-order model results at the same samples.
+The metric for comparison is the relative $L^2$ error defined as:
+
+\begin{equation}
+e = \frac{||T_{ROM}-T_{FOM}||}{||T_{FOM}||}
+\end{equation}
+
+This error is generated in a new input file, which computes the reconstructed field and
+solves the full-order model at the same time.
+
+!listing examples/stochastic/laser_welding_dimred/2d-reconst.i caption=Test input file for comparing the reconstructed fields to the reference solutions id=list:2d-reconst
+
+We see that duplication is minimized by reusing the same building blocks as the original input file.
+In this case, we load both the MOGP and the POD mapping from a file:
+
+!listing examples/stochastic/laser_welding_dimred/2d-reconst.i block=VariableMappings Surrogates
+
+And use them in an object that reconstructs the field variable and inserts it into a `T_reconst`
+auxiliary variable.
+
+!listing examples/stochastic/laser_welding_dimred/2d-reconst.i block=UserObjects
+
+Then, the integrated L2 error is computed by the appropriate postprocessor.
+
+!listing examples/stochastic/laser_welding_dimred/2d-reconst.i block=Postprocessors
+
+This input file is run for 90 samples of new process parameter combination selected by
+a Monte Carlo sampler. The input file for automating the process of error analysis is presented below.
+
+!listing examples/stochastic/laser_welding_dimred/test.i id=list:2d-reconst-pp caption=Input file for the automation of the testing phase.
+
+This simply gathers the relative $L^2$ errors from the subapplications and
+presents the results in a `json` file.
+The error histogram is presented below. We see that the maximum integrated $L^2$
+error is below 1.6% with the majority of the samples having error
+in the 0-0.2% interval.
+
+
+!plot histogram
+  id=test-results caption=Histogram of the L2 errors over the 90 test samples.
+  filename=examples/stochastic/laser_welding_dimred/gold/error_to_plot.csv
+  data=[{'x':'l2error', 'name':'L2 Error'}]
+  layout={'xaxis':{'type':'linear', 'title':'Relative L2 Error'},
+          'yaxis':{'type':'linear','title':'Frequency'}}
+
+
+
+

modules/combined/doc/content/modules/combined/examples/stm_thermomechanics.md

@@ -44,7 +44,7 @@ The problem of interest involves a steady-state thermomechanics model. The geome
 
 !row-end!
 
-!listing examples/stochastic/graphite_ring_thermomechanics.i caption=Thermomechanics model input file id=list:thermo
+!listing examples/stochastic/thermomech/graphite_ring_thermomechanics.i caption=Thermomechanics model input file id=list:thermo
 
 ### Uncertain Parameters
 
@@ -96,11 +96,11 @@ Using [latin hypercube sampling](LatinHypercubeSampler.md), the thermomechanics
 | Polynomial Chaos --- Training | 7,344 | 13.7 hr  |
 | Polynomial Chaos --- Evaluation | 100,000 | 6.8 s  |
 
-!listing combined/examples/stochastic/lhs_uniform.i id=list:lhs caption=Latin hypercube sampling and statistics input file
+!listing combined/examples/stochastic/thermomech/lhs_uniform.i id=list:lhs caption=Latin hypercube sampling and statistics input file
 
-!listing combined/examples/stochastic/poly_chaos_train_uniform.i id=list:train caption=Polynomial chaos training input file
+!listing combined/examples/stochastic/thermomech/poly_chaos_train_uniform.i id=list:train caption=Polynomial chaos training input file
 
-!listing combined/examples/stochastic/poly_chaos_uniform.i id=list:eval caption=Polynomial chaos evaluation input file
+!listing combined/examples/stochastic/thermomech/poly_chaos_uniform.i id=list:eval caption=Polynomial chaos evaluation input file
 
 ### Statistics
 

modules/combined/doc/content/modules/combined/tutorials/index.md

@@ -5,6 +5,7 @@ physics modules can be used together:
 
 - [Introduction to Coupled Thermo-Mechanical Modeling](combined/tutorials/introduction/index.md)
 - [Using Stochastic Tools with Multiphysics Models](combined/examples/stm_thermomechanics.md)
+- [Using dimensionality reduction and full-field reconstruction with multiphysics models](combined/examples/stm_laserwelding_dimred.md)
 
 # Solid Isotropic Material Penalization (SIMP) Topology Optimization
 

modules/combined/examples/stochastic/laser_welding_dimred/2d-reconst.i

@@ -0,0 +1,87 @@
+!include parameters.i
+
+!include mesh.i
+
+[Variables]
+  [vel]
+    family = LAGRANGE_VEC
+  []
+  [T]
+  []
+  [p]
+  []
+  [disp_x]
+  []
+  [disp_y]
+  []
+[]
+
+[AuxVariables]
+  [T_reconst]
+  []
+[]
+
+!include physics_objects.i
+
+[UserObjects]
+  [inverse]
+    type = InverseMapping
+    mapping = pod
+    variable_to_fill = "T_reconst"
+    variable_to_reconstruct = "T"
+    surrogate = mogp
+    parameters = '${R} ${power}'
+    execute_on = TIMESTEP_END
+  []
+[]
+
+[Surrogates]
+  [mogp]
+    type = GaussianProcessSurrogate
+    filename = "train_mogp_out_mogp.rd"
+  []
+[]
+
+[VariableMappings]
+  [pod]
+    type = PODMapping
+    filename = "train_pod_out_pod.rd"
+  []
+[]
+
+[Executioner]
+  type = Transient
+  end_time = ${endtime}
+  dtmin = 1e-10
+  dtmax = 1e-5
+  petsc_options_iname = '-pc_type -pc_factor_shift_type'
+  petsc_options_value = 'lu       NONZERO'
+  petsc_options = '-snes_converged_reason -ksp_converged_reason -options_left'
+  solve_type = 'NEWTON'
+  line_search = 'none'
+  nl_max_its = 16
+  l_max_its = 100
+  [TimeStepper]
+    type = IterationAdaptiveDT
+    optimal_iterations = 10
+    iteration_window = 4
+    dt = ${timestep}
+    linear_iteration_ratio = 1e6
+    growth_factor = 1.25
+  []
+[]
+
+[Postprocessors]
+  [l2error]
+    type = ElementL2Difference
+    variable = T
+    other_variable = T_reconst
+  []
+[]
+
+[Debug]
+  show_var_residual_norms = true
+[]
+
+[Postprocessors]
+[]

modules/combined/examples/stochastic/laser_welding_dimred/2d.i

@@ -0,0 +1,52 @@
+!include parameters.i
+
+!include mesh.i
+
+[Variables]
+  [vel]
+    family = LAGRANGE_VEC
+  []
+  [T]
+  []
+  [p]
+  []
+  [disp_x]
+  []
+  [disp_y]
+  []
+[]
+
+!include physics_objects.i
+
+[Executioner]
+  type = Transient
+  end_time = ${endtime}
+  dtmin = 1e-10
+  dtmax = 1e-5
+  petsc_options_iname = '-pc_type -pc_factor_shift_type'
+  petsc_options_value = 'lu       NONZERO'
+  petsc_options = '-snes_converged_reason -ksp_converged_reason -options_left'
+  solve_type = 'NEWTON'
+  line_search = 'none'
+  nl_max_its = 5
+  l_max_its = 100
+  [TimeStepper]
+    type = IterationAdaptiveDT
+    optimal_iterations = 5
+    iteration_window = 1
+    dt = ${timestep}
+    linear_iteration_ratio = 1e6
+    growth_factor = 1.25
+  []
+[]
+
+[Reporters]
+  [solution_storage]
+    type = SolutionContainer
+    execute_on = 'FINAL'
+  []
+[]
+
+[Debug]
+  show_var_residual_norms = true
+[]