Estimating 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, an Ensemble Kalman Filter is applied to the model. Linear time-varying (LTV) systems are also contemplated.
1. A method of estimating the temperature distribution inside a part manufactured by powder bed fusion, 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 that is used to control a process to manufacture the part and mitigate defects.
2. The method of claim 1 , further comprising:
reducing the number of ordinary differential equations in the set of ordinary differential equations via a change in variables of the ordinary differential equations as a linear combination of the set of variables, thus providing a transformed variable set prior to estimating the values of the transformed variable set; and
prior to outputting the estimate of the set of variables, computing the estimate of the set of variables by reversing the reduction of the number of ordinary differential equations.
3. The method of claim 2 , wherein the reducing comprises residualization.
4. The method of claim 2 , wherein the reducing comprises balanced truncation.
5. The method of claim 1 , wherein the discretizing comprises a finite element method.
6. The method of claim 1 , wherein the discretizing comprises a finite volume method.
7. The method of claim 1 , wherein the discretizing comprises a boundary element method.
8. A method of estimating the temperature distribution inside a part manufactured by powder bed fusion, 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;
constructing a set of copies of the set of ordinary differential equations with independent process and measurement noise distributions;
estimating the values of the variable set and noise distributions of process and temperature measurement by using an ensemble Kalman filter, using the set of ordinary differential equations, information on the heat applied, and a measurement of a temperature of the part during the process at specific locations; and
outputting an estimate of the set of variables that is used to control a process to manufacture the part and mitigate defects.
9. The method of claim 8 , further comprising:
prior to constructing the set of copies, reducing the number of ordinary differential equations in the set of ordinary differential equations via a change in variables of the ordinary differential equations as a linear combination of the set of variables, thus providing a transformed variable set; and
prior to outputting the estimate of the set of variables, computing the estimate of the set of variables by reversing the reduction of the number of ordinary differential equations.
10. The method of claim 9 , wherein the reducing comprises residualization.
11. The method of claim 9 , wherein the reducing comprises balanced truncation.
12. The method of claim 8 , wherein the discretizing comprises a finite element method.
13. The method of claim 8 , wherein the discretizing comprises a finite volume method.
14. The method of claim 8 , wherein the discretizing comprises a boundary element method.
15. A system for estimating the temperature distribution inside a part manufactured by powder bed fusion, the system comprising:
a processing module configured to:
discretize a geometry of the part 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; and
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
an output module configured to output an estimate of the set of variables that is used to control a process to manufacture the part and mitigate defects.
16. The system of claim 15 , wherein the processing module is further configured to:
reduce the number of ordinary differential equations in the set of ordinary differential equations via a change in variables of the ordinary differential equations as a linear combination of the set of variables, thus providing a transformed variable set prior to estimating the values of the transformed variable set, and
prior to outputting the estimate of the set of variables, compute the estimate of the set of variables by reversing the reduction of the number of ordinary differential equations.
17. The system of claim 15 , wherein the discretizing uses an adaptive mesh that changes over time.
18. The system of claim 15 , wherein the discretizing comprises a finite element method or a finite volume method.
19. The system of claim 15 , wherein the discretizing comprises a boundary element method.
20. The system of claim 15 , wherein the reducing comprises one of residualization or balanced truncation.