Model predictive control (MPC) for controlling internal temperature distributions within parts being manufactured via the powder bed fusion process
Estimation algorithms, methods, and systems are provided that estimate the internal temperatures inside of a part being built using powder bed fusion (PBF). Closed-loop state estimation is applied to the problem of monitoring temperature fields within parts during the PBF build process. A simplified linear time-invariant (LTI) model of PBF thermal physics with the properties of stability, controllability and observability is presented. In some aspects, Model Predictive Control (MPC) may be used as an expanded application of an Ensemble Kalman Filter (EnKF) methods to control the PBF process. MPC is used to forecast the PBF build process behavior N time steps into the future and identifies inputs that drive the temperature of a corresponding node in a mesh of n nodes towards a predetermined target temperature.
1. A method of estimating the temperature distribution inside a part manufactured by powder bed fusion process, the method comprising:
discretizing the part geometry into finite units to convert a thermal transport model for the part geometry into a set of ordinary differential equations that describes heat transport between elements, the discretizing defining a set of variables that represent temperature at specific spatial locations, using the thermal transport model and thermal properties of the part material as functions in the set of ordinary differential equations;
estimating the values of the variable set using a Kalman filter, using the set of ordinary differential equations, information on heat applied, an estimate of a noise distribution of the process, a measurement of a temperature of the part during the process at specific locations, and an estimate of a noise distribution in the temperature measurement; and
outputting an estimate of the set of variables to a model predictive control (MPC) to alter a set of control inputs to the power bed fusion process to drive the estimate to a predetermined target temperature field.
2. The method of claim 1 , wherein a linear time-invariant (LTI) process is used to alter the control inputs.
3. The method of claim 2 , wherein the LTI process model is described by:
x ( t )= Ax ( t )+ Bu ( t )
y ( t )= Cx ( t )
wherein u(t)∈ m represents heat applied to the exposed surfaces of all elements on face Ω in the mesh,
wherein Ω is the face of the part exposed to the laser (as described in the current application),
wherein y(t)∈ p collects the temperatures recorded in each pixel of an infrared (IR) camera with a fixed FOV that covers all of Ω, and
wherein A describes heat conduction between nodes in the FEM mesh, B maps the laser heat onto the relevant nodes in the mesh, and C models the mapping between nodal temperatures and measurements.
4. The method of claim 3 , the MPC further comprising:
declaring a first array X that stores forecasts for a process state over a forecast horizon of η time steps; and
declaring a second array U that stores forecasts for the control inputs over the forecast horizon.
5. The method of claim 4 , further comprising jointly optimizing both X and U such to conform to the predetermined target temperature field over the horizon.
6. The method of claim 4 , further comprising jointly optimizing both X and U such to conform to the predetermined target temperature field over the horizon.
7. The method of claim 1 , wherein a linear time-varying (LTV) process is used to alter the control inputs.
8. The method of claim 7 , wherein the LTV process model is defined by:
{dot over (x)} ( t )= Ax ( t )+ B ( t ) u ( t )
y ( t )= C ( t ) x ( t )
wherein components of u(t)∈ m are parameters of the laser beam, including beam power, radius, and position,
wherein y(t)∈ p collects temperatures recorded in each pixel of an infrared (IR) camera that has a field of view (FOV) is centered around the laser position and moves with time,
wherein A describes heat conduction between nodes in the FEM mesh, B(t) maps the laser heat onto the relevant nodes in the mesh, and because these parameters are time-varying, so too is this matrix time-varying, and C(t) models the mapping between nodal temperatures and measurements, which is time-varying because the camera FOV changes with time.
9. The method of claim 8 , the MPC further comprising:
declaring a first array X that stores forecasts for a process state over a forecast horizon of η time steps; and
declaring a second array U that stores forecasts for the control inputs over the forecast horizon.
10. The method of claim 1 , wherein components of the control input differ based on if a linear time-invariant (LTI) or linear time-varying (LTV) process model is used.
11. A system for estimating the temperature distribution inside a part manufactured by powder bed fusion, the system comprising:
a processing module configured to:
discretize the part geometry into finite units to convert a thermal transport model for the part geometry into a set of ordinary differential equations that describes heat transport between elements, the discretizing defining a set of variables that represent temperature at specific spatial locations, using the thermal transport model and thermal properties of the part material as functions in the set of ordinary differential equations;
estimate the values of the variable set using a Kalman filter, using the set of ordinary differential equations, information on heat applied, an estimate of a noise distribution of the process, a measurement of a temperature of the part during the process at specific locations, and an estimate of a noise distribution in the temperature measurement; and
output an estimate of the set of variables to a model predictive control (MPC) to alter a set of control inputs to the power bed fusion process to drive the estimate to a predetermined target temperature field.
12. The system of claim 11 , wherein a linear time-invariant (LTI) process is used to alter the control inputs.
13. The system of claim 12 , wherein the LTI process model is described by:
{dot over (x)} ( t )= Ax ( t )+ Bu ( t )
y ( t )= Cx ( t )
wherein u(t)∈ m represents heat applied to the exposed surfaces of all elements on face Ω in the mesh,
wherein Ω is the face of the part exposed to the laser (as described in the current application),
wherein y(t)∈ p collects the temperatures recorded in each pixel of an infrared (IR) camera with a fixed FOV that covers all of Ω, and
wherein A describes heat conduction between nodes in the FEM mesh, B maps the laser heat onto the relevant nodes in the mesh, and C models the mapping between nodal temperatures and measurements.
14. The system of claim 13 , the processing module further configured to:
declare a first array X that stores forecasts for a process state over a forecast horizon of η time steps; and
declare a second array U that stores forecasts for the control inputs over the forecast horizon.
15. The system of claim 14 , the processing module further configured to jointly optimize both X and U such to conform to the predetermined target temperature field over the horizon.
16. The system of claim 14 , the processing module further configured to jointly optimize both X and U such to conform to the predetermined target temperature field over the horizon.
17. The method of claim 11 , wherein a linear time-varying (LTV) process is used to alter the control inputs.
18. The system of claim 17 , wherein the LTV process model is defined by:
{dot over (x)} ( t )= Ax ( t )+ B ( t ) u ( t )
y ( t )= C ( t ) x ( t )
wherein components of u(t)∈ m are parameters of the laser beam, including beam power, radius, and position,
wherein y(t)∈ p collects temperatures recorded in each pixel of an infrared (IR) camera that has a field of view (FOV) is centered around the laser position and moves with time,
wherein A describes heat conduction between nodes in the FEM mesh, B(t) maps the laser heat onto the relevant nodes in the mesh, and because these parameters are time-varying, so too is this matrix time-varying, and C(t) models the mapping between nodal temperatures and measurements, which is time-varying because the camera FOV changes with time.
19. The system of claim 18 , the processing module further configured to:
declare a first array X that stores forecasts for a process state over a forecast horizon of η time steps; and
declare a second array U that stores forecasts for the control inputs over the forecast horizon.
20. The system of claim 11 , wherein components of the control input differ based on if a linear time-invariant (LTI) or linear time-varying (LTV) process model is used.