Attorney Docket No.2115-008409-WO-POA AN INTELLIGENT SCAN SEQUENCE OPTIMIZATION TECHNIQUE BASED ON THERMOMECHANICAL MODELS AND OBJECTIVES CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims the benefit of U.S. Provisional Application No. 63/635,749, filed on April 18, 2024. The entire disclosure of the above application is incorporated herein by reference. FIELD [0002] The present disclosure relates to intelligent scan sequence optimization techniques for laser powder bed fusion and, more particularly, relates to an intelligent scan sequence optimization technique based on thermomechanical models and objectives. BACKGROUND AND SUMMARY [0003] This section provides background information related to the present disclosure which is not necessarily prior art. This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features. [0004] Laser powder bed fusion (LPBF) is an increasingly popular approach for additive manufacturing (AM) of metals (and other materials). With reference to FIG.18, a laser powder bed fusion device 60 includes a bed of metal powder 62 on a build plate 64 and a concentrated laser beam apparatus 66 that is controlled by a controller. After each laser sequence, a new layer of metal powder is spread on the surface and the laser beam apparatus is again operated to melt the metal layer to create the part. It is used in various industries, ranging from aerospace, to automotive, and biomedical. It builds 3D parts by using a high-power laser to selectively fuse powder layer by layer. Compared with other AM techniques for metals, LPBF is popular for relatively high speed fabrication of complex parts with intricate features and high density at relatively high tolerances. However, parts produced by LPBF are prone to residual stress, deformation, cracks and other defects linked to high temperature gradients during the process. While post-process heat treatments can ameliorate come of these issues, they are often time-consuming, costly, and insufficient in rectifying deformation or cracks induced by thermal stresses during the build process. Consequently, it is critical to minimize temperature gradients during the build process to assure quality.
Attorney Docket No.2115-008409-WO-POA [0005] In LPBF, scan sequence (or scan strategy) refers to the order in which a pre- specified infill pattern is scanned. For example, the infill pattern shown in FIG.1a consists of a vector pattern of nine lines that could be scanned in different sequences. As an example, the so-called “alternating” scan sequence is shown by the ordering of the numbers in FIG.1a. By way of further example, the infill pattern shown in FIG.1b consists of an island pattern of nine islands could be scanned in different sequences. As an example, the so-called “island” scan sequence is shown by the ordering of the numbers in FIG.1b. Research has shown the importance of scan sequence in influencing macro-scale temperature distribution, residual stress, and shape distortion in parts built by LPBF. Despite its important role, scan sequence is currently selected via trial-and-error or heuristics, leading to suboptimal impacts on temperature distribution, residual stress and distortion. [0006] The present disclosure provides an intelligent scan sequence optimization technique for laser powder bed fusion and, more particularly, provides an intelligent scan sequence optimization technique based on thermomechanical models and objectives. [0007] Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. BRIEF DESCRIPTION OF THE DRAWINGS [0008] The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure. [0009] FIG.1a is an example vector pattern scan sequence for a laser powder bed fusion scan pattern of nine lines; [0010] FIG.1b is an example island pattern scan sequence for a laser powder bed fusion scan pattern of nine islands; [0011] FIG.2 is a block diagram of a smart scan sequence generation approach for laser powder bed fusion using an infrared camera and thermal models; [0012] FIG.3 is a simplified finite difference model for the laser powder bed fusion scan with thermomechanical model;
Attorney Docket No.2115-008409-WO-POA [0013] FIG.4 is an excerpt from preliminary work showing significantly improved thermal distribution and reduced deformation using a thermal model only scan technique (SmartScan 1.0) which is a predecessor to the present disclosure as compared to three common heuristic approaches for 2 dimensional scanning; [0014] FIG.5 shows preliminary results showing significantly reduced deflection using SmartScan 1.0compared to common heuristic approaches in printing a cantilever beam; [0015] FIG.6 is a graph of residual stresses for two heuristic approaches as compared to SmartScan 1.0; [0016] FIG.7 is a block diagram of a SmartScan 2.0 scan sequence generation approach for laser powder bed fusion using thermomechanical models (i.e. a combination of thermal and mechanical models) according to the principles of the present disclosure; [0017] FIG.8 is a schematic view of an example cantilever beam used in experimentation development of SmartScan 2.0; [0018] FIG.9 is a side view of a model of a cantilever beam; [0019] FIG.10 is a graph of simulated temperature and elastic deflection of cantilever beam for three scanning approaches; [0020] FIG.11 is an illustration of example cantilever beam deflection; [0021] FIG.12 illustrates the nodal points for calculating the temperature gradients; [0022] FIG.13 is a SmartScan 2.0 optimization workflow; [0023] FIG.14 illustrates case studies with vector and island scan patterns under three constraint scenarios: (a) Fixed left edge; (b) Fixed left and right edges; and (c) Fixed left and bottom edges; [0024] FIG.15 illustrate for case study 1: comparison of SmartScan 1.0 and 2.0 (applied to vector and island scan patterns) with regard to the: a) thermal uniformity metric Rf and (b) thermomechanical metric Jf , as functions of scanning progress; [0025] FIG.16 illustrate for case study 2: comparison of SmartScan 1.0 and 2.0 (applied to vector and island scan patterns) with regard to the: a) thermal uniformity metric Rf and (b) thermomechanical metric Jf , as functions of scanning progress; [0026] FIG.17 illustrate for case study 3: comparison of SmartScan 1.0 and 2.0 applied to vector and islaned scan patterns) with regard to the: a) thermal uniformity metric Rf and (b) thermomechanical metric Jf , as functions of scanning progress; and
Attorney Docket No.2115-008409-WO-POA [0027] FIG.18 is a schematic view of a laser powder bed fusion device according to the principals of the present disclosure. [0028] Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings. DETAILED DESCRIPTION [0029] Example embodiments will now be described more fully with reference to the accompanying drawings. [0030] Example embodiments are provided so that this disclosure will be thorough, and will fully convey the scope to those who are skilled in the art. Numerous specific details are set forth such as examples of specific components, devices, and methods, to provide a thorough understanding of embodiments of the present disclosure. It will be apparent to those skilled in the art that specific details need not be employed, that example embodiments may be embodied in many different forms and that neither should be construed to limit the scope of the disclosure. In some example embodiments, well-known processes, well-known device structures, and well-known technologies are not described in detail. [0031] According to the principles of the present teachings, an intelligent approach for optimally selecting a laser powder bed fusion scan sequence that utilizes both thermal and mechanical models (i.e. SmartScan 2.0). With reference to FIG.18, a laser powder bed fusion device 60 includes a metal powder bed 62 on a build plate 64 and a laser device 66 for melting the metal powder at select locations as controlled by a controller. The powder bed 62 is relayered after each scan sequence is completed to create a new powder layer that is agains scanned with the laser to continue to build each layer. The fabricated part is supported on the build plate 64 that can be subsequently removed. [0032] As shown in FIG.2 the SmartScan 1.0 process 10 combines physics based thermal models 12 with the thermal history data 14 obtained using an infrared (IR) camera 16 to determine the optimal sequence for scanning the next layer. An optimization process 18 is completed within the interlayer time during which the powder is being spread for the next layer–typically lasting about 10-40 seconds. [0033] Testing of the SmartScan 1.0 process used only a physics-based thermal model is discussed below. For this purpose, a simplified finite difference method (FDM) thermal model was used. For simplicity, a single layer of LPBF on a substrate was assumed. Only conductive and convective modes of heat transfer were assumed to
Attorney Docket No.2115-008409-WO-POA occur between the layer and substrate, or between the layer, substrate and their surroundings (see Fig.3). Radiative heat transfer, latent heat effects, Marangoni convection, and other melt pool phenomena, were ignored. Based on the FDM model, the statespace equation of Eq. (1) can be written: ^(^ + 1) = ^^(^) + ^^(^) (1) where T(l) is the state vector comprising the T of all elements at time l; A is the state matrix; B is the input matrix; and u(l)
the power input to each element at time l. The boundary conditions are embedded in matrix A. If it takes np time steps of length Δt to scan each feature (e.g., vector or island), the temperature distribution resulting from scanning each feature can be expressed as Eq. (2): ^
[0034] Note that the feature-level state equation given by Eq. (2) has a sampling interval of Δtp = npΔt. A thermal uniformity metric R(lp) was used to evaluate the temperature distribution across the layer after each feature is scanned. It is defined as in Eq. (3):
where Tavg(lp) is the average of the temperatures T(i,j,k,lp) of all ne elements of the layer at time lp, and Tm is the melting temperature of the material. A lower value of R(lp) implies more uniform temperature distribution. Let the goal of the optimal scan sequence be to minimize the temperature uniformity metric (defined in Eq. (3)) across any given layer. This objective could be expressed as in Eq. (4),
where Aeq = Ap; while the columns of Beq represent the bp vectors corresponding to each feature. In Ceq, I is the identity matrix, 1 is a column vector whose elements are all equal to 1, and 0 is a null matrix used to account for any elements of T(lp) that are not needed to calculate R(lp). The vector ueq(lp) consists of one element equal to 1 and all other elements equal to 0. The location of 1 in ueq(lp) represents the column of Beq to be selected (i.e.: the location of the feature to be scanned) to optimize the temperature distribution T(lp+1) by minimizing R(lp+1) at the next time step lp +1. The argument of the objective of Eq. (4) can be re-written using the square of R(lp+1) as shown in Eq. (5):
Attorney Docket No.2115-008409-WO-POA 2 2 2 !^^^^ + 1^' = ./01^^^^ + 1^.2 = ./01^01^^^^^ + /01^01^01^^^^.2 ; ; ; (5) = ^ ^ ; ; ; ; ; ; 01 ^^^^01 /01 /01^01^01^^^^ + 2^ ^^^^^01 /01 /01^01^01^^^^ + ^ ^^^^^01 /01
[0035] The last term of Eq. (5) is not affected by scan sequence ueq(lp). Also, the structure ueq(lp), consisting of only ones and zeros, makes only the diagonal terms of the matrix Beq T Ceq T CeqBeq important for the optimization. This allows the optimization problem of Eq. (4) to be simplified to Eq. (6): (6) ^ #+, = s.t. > = diagD^D^D; D /E; D /DDD^DF^ + 2C^DD; D /D; E /DDD^DF I J; # CD 01 01 01 01 01 01 01 01 ^^^^^ where > = =# (6) G H [0036] Since Γ and Λ in Eq. (6) are known a priori, they can be computed offline. Also, the initial temperature distribution T(0) is known. Therefore, starting from time lp = 0, determining the optimal scan sequence, at any given time step lp, becomes as simple as: (I) calculate the vector γ from Eq. (6) and determine the index i corresponding to its smallest element; (II) select the optimal feature at time lp that minimizes R(lp+1) by entering 1 as the element of ueq(lp) corresponding to index i; (III) calculate the optimal thermal distribution T(lp+1) using the state equation in Eq. (4); (IV) advance to time lp+1 and repeat the process from (I) until the entire layer is scanned. Note that the point-to-point positioning time of the laser is not included in the formulation above because it is negligible compared to the time spent scanning. Furthermore, radial basis functions are employed for model-order reduction in Eq. (6) as discussed in Ramani, K. S., He, C., Tsai, Y. L., & Okwudire, C. E. (2022). SmartScan: An Intelligent Scanning Approach for Uniform Thermal Distribution, Reduced Residual Stresses and Deformations in PBF Additive Manufacturing. Additive Manufacturing, 102643. This improves computational efficiency with negligible reduction in optimality. [0037] The simplified knowledge-driven approach described above was applied to laser marking of unconstrained 316L stainless steel plates in experiments on a PANDA LPBF machine using vector and island patterns. For the sake of brevity, the details of the experiments are omitted here but can be found in Ramani, K. S., He, C., Tsai, Y. L., & Okwudire, C. E. (2022). SmartScan: An Intelligent Scanning Approach for Uniform Thermal Distribution, Reduced Residual Stresses and Deformations in PBF Additive Manufacturing. Additive Manufacturing, 102643.. A few highlights of the results for the island scan pattern are as follows: [0038] SmartScan 1.0 outperformed three state-of-the-art heuristic methods (Successive, Successive Chessboard and LHI, explained in Sec.3.2). It yielded up to
Attorney Docket No.2115-008409-WO-POA 41% improvement in thermal uniformity (R) and up to 47% reduction in thermal- induced distortion (see Fig.5). [0039] SmartScan 1.0 computed very quickly (in 6 seconds) which falls within the interlayer time of LPBFand makes it suitable for future online implementation. [0040] SmartScan 1.0 took the same time as the heuristic methods to scan each island. However, it required at most 2.7% more time for longer jumps from island to island to follow the optimal sequence. [0041] Similar results were obtained for the vector pattern in Ramani, K. S., He, C., Tsai, Y. L., & Okwudire, C. E. (2022). SmartScan: An Intelligent Scanning Approach for Uniform Thermal Distribution, Reduced Residual Stresses and Deformations in PBF Additive Manufacturing. Additive Manufacturing, 102643, but are omitted here for the sake of brevity. [0042] Further work has expanded the single-layer formulation presented above for the 2D case to 3D, thus allowing parts like a cantilever beam 20 shown in FIG.5(a) to be printed. FIG.5(b) shows the deformed beams after printing and sawing the support structures off the build plate and FIG.5(c) shows laser scans of the top surfaces of the printed parts. SmartScan 1.0 exhibits up to 1.6x lower peak deformation compared to state-of-the-art heuristic methods like <sequential,= where the pattern is followed sequentially and alternating, which is described in Fig.1. Fig.6 shows the residual stress in the x direction measured at the point marked with the red dot on Fig.5(a), using a Rigaku SmartLab X-ray diffraction (XRD) machine. It is noted that SmartScan 1.0 yields up to 9x lower residual stress than the heuristic methods. Beyond stripe and island patterns, SmartScan 1.0 has also been applied to advanced scan patterns, like the varying-helix pattern [34] with excellent results. [0043] The original SmartScan 1.0 algorithm depended on only thermal models and was based on optimizing a thermal uniformity metric given by Eq. (3). It is well understood that achieving thermal uniformity does not necessarily yield the best reduction in part distortion or residual stress. Therefore, the SmartScan 2.0 of the present process seeks to use thermomechanical models of LPBF to optimize objective functions that are directly connected to residual stress and part distortion. Fig.7 gives an overview of the SmartScan 2.0 with thermomechanical modeling. [0044] One condition where SmartScan 1.0 yields poor results is when distortion or residual stress is strongly dependent on mechanical boundary conditions. For example, consider the cantilever beam shown in FIGS.8, 9, which could represent an
Attorney Docket No.2115-008409-WO-POA unsupported overhang jotting out of a part being printed using LPBF. The top surface of the beam is marked with a laser using vectors numbered from 1 to 325. The cantilever beam experiences bending moment M(x) at point x given by Eq. (7), K = MNO("P(Q)^"R(Q)) [0045] where E is the beam’s
the beam’s 2nd moment of area, α is the coefficient of thermal expansion, H is the height of the beam, while Tb and Tt are the temperatures of the top and bottom surfaces of the beam, respectively. The ends of the beam are insulated while the top and bottom surfaces experience convection to the ambient temperature, Ta. [0046] The maximum elastic deflection T^UQ at x = L due to M(x) is given by Eq. (8). V
[0047] It is noted that T^UQ is proportional to (L−x)2. This means that the moments applied closer to the fixed end (x = 0) have a lot more influence on ^max than those applied near the free end (x = L). From Eq. (7), we can see that M(x) is proportional to Tt−Tb. If the beam is initially at a uniform temperature T0, M(x) is likely to have its largest value when the first vector is scanned because Tb = T0 and Tt = Tm (the melting temperature). As the scan progresses, the value of M(x) should gradually decrease as Tb increases due to heat conduction. Therefore, a heuristic scan sequence that starts at the free end of the beam to its fixed end (exploiting Eq. (8)) is more likely to yield a lower ·max than a SmartScan that jumps around the surface of the beam without exploiting Eq. (8). We assume that a scan sequence that leads to a larger T^UQ when only elastic deformation is considered (as in Eq. (8)) will lead to a larger plastic deformation in reality. [0048] For example, the cantilever beam of Fig.8 is modeled using a beam FEA mechanical model superimposed on a thermal finite difference model (FDM), as shown in Fig.9. T(i) = [Tt(i), Tb(i)] represents the temperatures of the top and bottom nodes of the FDM while D(i) = [T (i), θ(i)] represents the bending displacement and rotation of each cross section i. The scanning of the beam results in force vector
F(i) = [V(i), M(i)], where the shear force V(i) = 0 and the bending moment M(i) is given by Eq. (8) with x replaced by i. Since the relationship between F(i) and T(i) is linear, it can be written in matrix form as F = ΨT (where F and T are vectors composed of F(i) and T(i) for all cross sections i of the beam).
Attorney Docket No.2115-008409-WO-POA 2
^ ^+, !Y^^^ + 1^ = ./01^^^^ + 1^. )-^&^^ 2' s.t. ^^^^ + 1^ = ^01^^^^^ + ^01^01^^^^ (10) where / = ^^ 01 diag([1,0,1,0 … ])_ Ψ [0049] Assuming linear elastic behavior, D = K-1F. Therefore, D = K-1ΨT, where K is the stiffness matrix of the beam. Therefore, the thermal objective R of Eq. (3) can be modified to Q in Eq. (9) which represents the aggregate of the deflection at every cross section i of the beam. This objective can be applied to the optimization problem of Eq. (10) which has exactly the same structure of SmartScan 1.0 without the thermomechanical model given in Eq. (4). Therefore, it can be solved following the procedure by simply substituting Ċeq for Crq in Eq. (6). To experimentally test our hypothesis under this assumption, the heuristic sequence 325, 324, 323…3, 2, 1 (starting from the free end to the fixed end of the beam) is compared to SmartScan without the thermomechanical model whose first 10 sequences are 315, 296, 276, 256, 234, 214, 194, 174, 154 and 134, and SmartScan with thermomechanical models whose first 10 sequences are 325, 320, 318, 324, 322, 316, 323, 313, 312 and 321. [0050] It is noted that the SmartScan 2.0 with thermomechanical models is similar to but not exactly the same as the heuristic sequence in that it tends to move from the free end towards the fixed end of the beam. The experiments are performed on a PANDA 11 LPBF machine. FIGS.7 and 8 show the experimental set up using a 316 L stainless steel beam of dimensions L = 65 mm, W = 5 mm, H = 2 mm. A laser power of 300 W and scan speed of 1200 mm/s are used to mark its top surface. [0051] Fig.10 shows the simulated temperature distributions and elastic beam deflections of the heuristic, SmartScan 1.0 without the thermomechanical model and SmartScan 2.0 with the thermomechanical model sequences at 25%, 50%, 75% and 100% of the scanning process. Fig.11 shows the scanned beams. SmartScan 2.0 with the thermomechanical model yields 49% and 16% lower maximum deflection than SmartScan 1.0 without the thermomechanical model and the heuristic approach, respectively. This example shows the potential benefits of incorporating SmartScan 2.0 with the thermomechanical model.
Attorney Docket No.2115-008409-WO-POA [0052] It is anticipated that the SmartScan 2.0 process can use more advanced thermomechanical models (e.g., advanced FEA models) to generate general mappings between temperature (T) and part deformation (D) and stress (S). One possible form of the mappings is D = ΦT or S = ΨT. Where Φ and Ψ are the mappings.
[0053] The SmartScan 2.0 process can define functions that directly optimize distortion or residual stress related objectives. For example, the process can minimize the maximum deformation or stress. The process can also minimize the total deformation or stress. The SmartScan 2.0 process can also leverage mathematical tools like linear systems theory and control theory to facilitate optimization. [0054] The present disclosure has demonstrated one way the three steps above can be achieved. Moreover, the proposed method could be applied beyond LPBF. It could be applied to electron beam powder bed fusion, directed energy deposition methods, welding, and other manufacturing processes. [0055] The SmartScan 2.0 process can be implemented to directly embed simplified thermomechanical models into optimal scan sequences using finite element analysis (FEA). The goal of thermomechanical modeling is to determine key indicators from the thermal and mechanical models of LPBF to inform the objective function for SmartScan 2.0. The thermal model is used to select the local temperature gradient at the laser’s location as the thermal indicator for scan sequence optimization. Then, the mechanical model is used to select the local compliance at the laser’s location as the mechanical indicator for scan sequence optimization. In both cases, the models are simplified (linearized and discretized) to ensure that they are amenable to efficient scan sequence optimization. Fig.7 gives an overview of the SmartScan 2.0 with thermomechanical modeling. THERMAL MODELING TO EXTRACT THERMAL INDICATOR FOR SCAN SEQUENCE OPTIMIZATION [0056] According to the principles of the present teachings, an intelligent approach for optimally selecting a laser powder bed fusion scan sequence that considers the physics of the problem, as well as thermal data gathered from the machine during the printing process, is provided (i.e. SmartScan). As shown in FIG.7 the SmartScan 2.0 with thermomechanical process 40 combines physics based thermal models 42 obtained using an infrared (IR) camera 46 combined with mechanical models 44 to determine the optimal sequence for scanning the next layer. An optimization process 48 is completed
Attorney Docket No.2115-008409-WO-POA within the interlayer time during which the powder is being spread for the next layer– typically lasting about 10-40 seconds. [0057] For the thermal model, the model considers conductive and convective heat transfer between the part and its surroundings while making several simplifying assumptions. Specifically, radiative heat transfer, latent heat effects, Marangoni convection, and other melt pool phenomena are ignored. Temperature-independent model parameters, resulting in a linear model suitable for efficient optimization are assumed. Under these assumptions, heat transfer within the printed part with conductivity kt and diffusivity α is governed by the equation: (" (" (" ^
where T represents temperature, (x, y, z) are spatial coordinates, t is time, and u denotes power per unit volume. The equation can be discretized using the finite difference method (FDM) to yield: "
where i, j, k are spatial indices for the elements of dimensions Δx, Δy, Δz, respectively, and l represents the temporal index for time step Δt. The temperature at position i, j, k and time step l is denoted as T(i, j, k, l). The simplified FDM model can be expressed as a state equation: ^(^ + 1) = ^^(^) + ^^(^) (13) where T(l) is the state vector containing the temperatures of all ne elements at time step l, A is the state matrix, B is the input matrix, and u(l) represents the power input vector that has nonzero values only at the elements actively heated by the laser at any given time. The model incorporates convection to ambient temperature Ta, with coefficient h, on its top and bottom surfaces. The side edges are considered to be insulated, since a plate is assumed for this initial investigation, and the heat transfer from its side edges are negligible due to their small surface areas. [0058] Recall that the features of a scan pattern are its repeating geometric units (e.g., vectors or islands). For a total of nF features, let feature f require Nf time steps for laser scanning, with its time steps recorded in a set gf = {l f,0, l f,1, ..., l f,Nf −1}. When scanning
Attorney Docket No.2115-008409-WO-POA each feature f ∈ {1, 2, ..., nF}, the temperature at any time point l f,m can be expressed as: = + where l
f,0 denotes , of time steps that have elapsed at the point of interest. Note that the thermal model assumes zero transition time between consecutive features (i.e., it assumes that the jump speed is infinite) for the sake of simplicity. This assumption makes sense because the jump speed is typically much larger than the scanning speed. [0059] Prior studies have identified local temperature gradients as primary contributors to residual stress and deformation in LPBF. The rapid heating and cooling cycles during LPBF generate steep thermal gradients, resulting in differential thermal expansion and contraction. These thermomechanical interactions ultimately manifest as residual stress and part deformation. Acccordingly, the thermal indicator IT is defined as the L1 norm of the temperature gradient at point L located on feature f being scanned at time step l f,m. It is given by:
where (iL, jL, kL, l f,m) represents the point (iL, jL, kL) being heated by the laser at time step l f,m. Using the centered average difference method, IT can be rewritten in discrete- form as: k
gc [0060] Note that in the discretized expression for IT given in Eq.16, the absolute value in the L1 norm of Eq.15 has been eliminated. This is because the temperature gradient at (iL, jL, kL) is always positive, since point L is always at a higher temperature than its surrounding points. This makes IT a linear function with respect to temperature T, thereby facilitating efficient optimization. Notice also that the mean of differences on both sides of the laser application point (iL, jL, kL) is computed for the x and y directions. However, since the laser is always applied on the top surface, the temperature gradient along z axis only considers the downward direction (See Fig.12).
Attorney Docket No.2115-008409-WO-POA MECHANICAL MODELING TO EXTRACT MECHANICAL INDICATOR FOR SCAN SEQUENCE OPTIMIZATION [0061] The part is modeled using finite element analysis (FEA). For this initial investigation, without loss of generality, a custom FEA model was developed following the standard methodology detailed in Bhavikatti S. Finite element analysis. New Age International; 2005. The part, assumed to be a plate in this paper, is discretized using 8-node hexahedral elements of dimensions Δx, Δy, Δz, which match the dimensions and number of elements in the thermal FDM model of (Fig.3). The mechanical boundary conditions are applied using standard FEA techniques. For mathematical convenience, the following simplifications are adopted aimed at keeping the FEA model linear to facilitate efficient optimization: Constant material properties (e.g., temperature- independent Young’s modulus and Poisson’s ratio), purely elastic deformation, and small strain assumptions. [0062] To determine the mechanical indicator, we observe from the literature for laser forming that boundary conditions and their effects on local deformation significantly affect the mechanical response during laser scanning [Birnbaum AJ, Cheng P, Yao YL. Effects of clamping on the laser forming process. Journal of Manufacturing Science and Engineering.2007;129:1035-44; Cheng J, Yao YL. Cooling effects in multiscan laser forming. Journal of Manufacturing Processes.2001;3(1):60-72; [47] Edwardson S, Griffiths J, Edwards K, Dearden G, Watkins K. Laser forming: overview of the controlling factors in the temperature gradient mechanism. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science.2010;224(5):1031-40.]. For example, it has been shown that reductions in part distortion can be achieved by initiating scanning from regions far from high-constraint (stiffer) areas to allow freer thermal and stress relief in the more compliant regions far from the constraints [Birnbaum AJ, Cheng P, Yao YL. Effects of clamping on the laser forming process. Journal of Manufacturing Science and Engineering.2007;129:1035- 44]. Conversely, scanning near constrained regions first can result in greater part distortion and stress due to thermal and stress accumulation in the stiffer areas [Birnbaum AJ, Cheng P, Yao YL. Effects of clamping on the laser forming process. Journal of Manufacturing Science and Engineering.2007;129:1035-44]. Therefore, we seek to define a mechanical indicator that captures the effects of constraints and mechanical stiffness/compliance on scanning performance.
Attorney Docket No.2115-008409-WO-POA [0063] To do this, we extract the global stiffness matrix K ∈ R3ne×3ne from the part’s FEA model. Using this information, we determine the part’s compliance matrix C ∈ R3ne×3ne as C = K−1. Each element cpq of C represents the displacement at the p-th degree of freedom under a unit load applied at the q-th degree of freedom. The displacement response norm D(i, j, k) for each node (i, j, k) can be calculated from submatrix Cijk, comprising the three rows corresponding to the node’s x, y, and z translational degrees of freedom (i,e, rows 3n − 2 through 3n, where n is the node number). It is defined using the Frobenius norm as:
[0064] This scalar measure D(i, j, k) characterizes the node’s aggregate displacement under all possible loading conditions, with larger values indicating less constrained nodes. To manage the wide range of displacement response norms, we apply a logarithmic transformation to D(i, j, k), yielding: s&vw(+, m, n) = −log^^ (s(+, m, n)) (18) [0065] This transformation compresses the range of D(i, j, k) while maintaining the relative differences in displacements among the nodes. Finally, we determine the mechanical indicator IM(iL, jL, kL, lf,m) at each location L being scanned by the laser at time step l f,m by normalizing Dlog(iL, jL, kL) within the interval [1, 5] using a linear scaling as:
where max (Dlog (I,j,k)) and min (Dlog (I,j,k)) represent the global extrema of Dlog(i, j, k) over all the nodes in the part. In this normalized scale, IM = 1 represents the lowest compliance level (typically at fixed supports), while IM = 5 indicates the highest compliance level (usually at free ends). [0066] The normalization of IM(iL, jL, kL, lf,m) within the interval is arbitrary. A different range can be chosen as long as it preserves the relative compliance of the nodes in the part.
Attorney Docket No.2115-008409-WO-POA OBJECTIVE FUNCTION AND OPTIMIZATION METHODOLOGY [0067] The thermal indicator defined in the preceding section provides the fundamental driver for residual stress and distortion at each location L of the laser, while the mechanical indicator represents the effect of mechanical constraints on location L. Since the goal of SmartScan 2.0 with thermomechanical modeling is to select the best feature to be scanned at any given instance, the time average of the product of the mechanical indicator and thermal indicator over each feature f are selected as the objective function. This can be represented as: ^ ^ ^ ^^
[0068] The by computing the weighted average of the L1 norm of the thermal gradients across all points L along the scanning path of feature f , with weights determined by the local mechanical indicators at each point L. [0069] Since the thermal indicator is linear with respect to temperature and the mechanical indicator is a scalar constant, Jf can be rewritten as a linear operator given by:
where each column of Tf contains the global temperature field at a specific instance during the scanning of feature f. Correspondingly, each row of Φf extracts the L1 norm of the temperature gradient at the laser position L from the global temperature field to calculate the thermal indicator. Then, Ψf weights the thermal indicators by their corresponding mechanical indicators, normalized by Nf. Finally, the matrix operations compute the temporal average of the product of both indicators across the feature f to yield Jf . As such, the optimization problem takes the form:
Attorney Docket No.2115-008409-WO-POA arg ^i+, ^i ^ (22) s.t. ^i = diag^^i^i^ ^i
where Φf and Ψf can be pre-computed for each feature f ∈ {1, 2, ..., nF} and the value of f that minimizes Jf can be selected from the results, based on the value of Tf calculated just after the last feature is scanned. Fig.11 illustrates the SmartScan 2.0 optimization workflow. [0070] Note that the objective function Jf used in SmartScan 2.0 with thermomechanical modeling is very different from that used in SmartScan 1.0 with just thermal modeling, which was focused on optimizing the thermal uniformity metric, Rf , defined as: '^
[0071] where Tavg is the average of temperature T of all nodes at time l f,Nf −1, and Tm is the material’s melting temperature used as a normalization constant along with √ne. [0072] The optimization methodology for SmartScan 2.0 with thermomechanical modeling is very similar to that of SmartScan 1.0 with thermal modeling, detailed herein. All matrices (except Tf ) needed to determine the optimal solution can be pre- computed. This facilitates the computational efficiency of both algorithms. [0073] Simulation case studies are used to evaluate SmartScan 2.0’s performance under three mechanical constraint scenarios applied to a plate, as shown in Fig.14: • Case Study 1: Left edge fully constrained • Case Study 2: Left and right edges fully constrained • Case Study 3: Left and bottom edges fully constrained [0074] For each constraint configuration, we conduct comparative analyses between SmartScan 1.0 with thermal modeling and SmartScan 2.0 with thermomechanical modeling, involving vector and island scan patterns (see Fig.14). SIMULATION SETUP [0075] All simulations involve laser marking of AISI 316L stainless steel plates (50 mm×50 mm×1 mm) using the parameters listed in Table 1. The machine and process parameters are based on the PANDA 11 LPBF machine discussed in the experimentla setup section below. For the vector scan pattern, each plate is filled with 250 bidirectional vertical lines spaced at 200 μm intervals. For the island scan pattern, each
Attorney Docket No.2115-008409-WO-POA plate is partitioned into 100 islands, each of size 5 mm×5 mm. Each island is filled with 25 bidirectional vectors spaced at 200 μm that are rotated by 90 degrees between adjacent islands (e.g., see Fig.1(b)). The FDM simulation uses elements of size Δx = Δy = Δz = 200 μm and time step Δt = 0.333 ms. Convection boundary conditions are applied to the top and bottom surfaces, while the peripheral surfaces are considered to be adiabatic due to their negligible areas. The FEA model uses the same size and number of elements as the FDM model (i.e., Δx = Δy = Δz = 200 μm). Each FEA element is an eight-node hexahedral with trilinear shape functions and Gaussian quadrature for numerical integration. The global stiffness matrix K is constructed assuming linear elastic behavior with constant material properties, based on standard FEA procedures using the relevant parameters from Table 1. Note that even though the jump speed is listed in the table, for simplicity, the simulations assume infinite jump speed (i.e., instantaneous jumps) between consecutive features, since the jump speed is much higher than the mark/scan speed. [0076] For each simulation case study, we evaluate SmartScan 2.0 with thermomechanical models against SmartScan 1.0 with thermal models with regard to: (1) the scan sequence; (2) the thermal uniformity metric, Rf, which SmartScan 1.0 optimizes; (3) the thermomechanical objective, Jf, which SmartScan 2.0 with thermomechanical models optimizes, and (4) the simulated temperature distribution. Finally, the computation time for both methods are compared for all case studies. TABLE 1. Parameters used for simulations and experiments
Initial temperature, T(x, y, z, 0) (K) 293
Attorney Docket No.2115-008409-WO-POA Ambient temperature, Ta (K) 293
SIMULATION CASE STUDY 1: LEFT EDGE FULLY CONSTRAINED [0077] For both vector and island patterns, SmartScan 1.0 with thermal modeling jumps around the plate to distribute the temperature as evenly as possible. In contrast, SmartScan 2.0 with thermomechanical modeling generates a predominantly right-to-left scan sequence with some pre- and post-heating at the frontier of the motion from right to left to help minimize the local thermal gradients for each feature, as informed by the thermal indicator. The reason for the right-to-left direction of the scan sequences is because it is guided by the mechanical indicator to travel from the more compliant to the less compliant regions of the plate. [0078] In terms of Rf and Jf (see Fig.15), SmartScan 1.0 with thermal models demonstrates superior performance in terms of thermal uniformity, achieving up to a 32.8% reduction in the mean value of Rf compared to SmartScan 2.0 with thermomechanical modeling. Conversely, SmartScan 2.0 with thermomechanical modeling achieves up to a 32.3% reduction in the mean value of Jf compared to SmartScan 1.0 with thermal modeling, which bodes well for the thermomechanical deformation of the plates evaluated in the experiments. The large spikes in the Jf values of SmartScan 1.0 with thermal modeling are due to sudden increases in local thermal gradients that occur as the laser heats cold regions of the plate. SmartScan 2.0 with thermomechanical modeling avoids such spikes by performing local pre- and post- heating around each feature to minimize the local thermal gradients. [0079] The comparative performance of both methods with regard to temperature uniformity is corroborated by the simulated temperature distribution of the plates at four key stages (25%, 50%, 75%, and 100%) of the scanning process. It is apparent that SmartScan 1.0 with thermal modeling results in a more uniform temperature distribution than SmartScan 2.0 with thermomechanical modeling. SIMULATION CASE STUDY 2: LEFT AND RIGHT EDGES FULLY CONSTRAINED [0080] For the scan sequences for both vector and island patterns for Case Study 2, SmartScan 1.0 with thermal modeling jumps around the plate to distribute the temperature as evenly as possible. In contrast, SmartScan 2.0 with thermomechanical
Attorney Docket No.2115-008409-WO-POA modeling generates a sequence that predominantly travels from around the middle of the plate towards its left and right edges. As before, it also incorporates some pre- and post-heating at the frontier of the motion to minimize the local thermal gradients for each feature, as informed by the thermal indicator. The SmartScan 2.0 with thermomechanical sequence travels from the middle to the edges in order to scan the more compliant regions before the less compliant regions of the plate. Notice that while for the vector pattern SmartScan 2.0 with thermomechanical modeling starts scanning from the left of the centerline of the plate, for the island pattern it starts scanning from the right of the centerline. This may mean that, due to the symmetry of the constraints, the algorithm is agnostic to which side of the centerline it starts scanning. [0081] In terms of Rf and Jf (see Fig.16), again, SmartScan 1.0 with thermal modeling demonstrates superior performance in terms of thermal uniformity, achieving up to a 28.3% reduction in the mean value of Rf compared to SmartScan 2.0 with thermomechanical modeling. Conversely, SmartScan 2.0 with thermomechanical modeling achieves up to a 31.8% reduction in the mean value of Jf compared to SmartScan 1.0 with thermal modeling, which bodes well for the thermomechanical deformation of the plates evaluated in the experiments. SIMULATION CASE STUDY 3: LEFT AND BOTTOM EDGES FULLY CONSTRAINED [0082] For the scan sequences for both vector and island patterns for Case Study 3, SmartScan 1.0 with thermal modeling follows the same pattern as with Case Studies 1 and 2, because it cannot account for mechanical boundary conditions. However, for the vector pattern, SmartScan 2.0 with thermomechanical modeling generates a sequence that predominantly travels from the right to the left edges of the plate, while for the island pattern, the sequence travels from the top right corner towards the bottom left corner of the plate. As before, both cases incorporate some pre- and post-heating at the frontier of the motion to minimize the local thermal gradients. Ideally, the SmartScan 2.0 with thermomechanical modeling sequence for the vector pattern would follow the same trend as that of the island (as informed by the mechanical indicator) but it cannot do so because it lacks the degrees of freedom to travel diagonally. Therefore, it does the best it can with its available degrees of freedom by traveling from right to left. In terms of Rf and Jf (see Fig.17), again, SmartScan 1.0 with thermal modeling demonstrates superior performance in terms of thermal uniformity, achieving up to a 32.8% reduction in the mean value of Rf compared to SmartScan 2.0. Conversely,
Attorney Docket No.2115-008409-WO-POA SmartScan 2.0 with thermomechanical modeling achieves up to a 32.2% reduction in the mean value of Jf compared to SmartScan 1.0 with thermal modeling. COMPARISON OF THE COMPUTATIONAL EFFICIENCIES OF SMARTSCAN 1.0 AND 2.0 [0083] The computational efficiency of the SmartScan 2.0 algorithms is very important. For real-time implementation, it is important for the optimization of each layer to be completed in less than the interlayer time of LPBF during which powder is recoated. In our prior work, SmartScan 1.0 with thermal modeling has been shown to complete the optimization of each layer in less than 15 seconds, which falls within the typical interlayer time of LPBF. Table 2 shows that both SmartScan 1.0 with thermal modeling and SmartScan 2.0 with thermomechanical modeling meet the <15 seconds criterion in all cases. TABLE 2. Computation time of SmartScan 1.0 and 2.0 for the three cases studied ) I I N
9 7945HX processor, 32.0 GB RAM and NVIDIA GeForce RTX 4070 Laptop GPU. EXPERIMENTAL SETUP AND PROCEDURE [0084] Case studies 1 through 3 from Section 3 are experimentally validated in this section using an open-architecture PANDA 11 LPBF machine (OpenAdditive, LLC, Beavercreek, OH), equipped with a 500W IPG Photonics 1070 nm fiber laser and a SCANLAB hurrySCAN galvo scanner with an f-theta lens. The machine, controlled by the Open Machine Control software, enables custom scan patterns and sequences to be implemented. A nitrogen gas cross-flow arrangement at 0.5-1.5 L/min was maintained during the experiments. The primary goal of the experiments was to assess whether the reductions in Jf achieved using SmartScan 2.0 with thermomechanical modeling compared to SmartScan 1.0 with thermal modeling lead to any significant
Attorney Docket No.2115-008409-WO-POA improvements in the thermomechanical performance (in this case, the deformation) of the final parts. The experiments involve marking a roughly 50 mm × 50 mm area on AISI 316L stainless steel plates (50 mm × 50 mm × 1 mm). Each plate was supported by a 3D-printed PLA frame (thermal conductivity 0.13 W/m・K) to minimize conductive heat transfer to the build plate, matching our simulation conditions. To constrain the edges, the frame was furnished with grooves laden with an adhesive to ensure proper constraints were applied. All process parameters follow Table 1 specifications. [0085] After scanning each plate using the method being tested, each plate was removed from the fixture and its thermomechanical deformation was measured using a Romer Absolute Arm 3D scanner (Hexagon, Stockholm, Sweden, model #7525SI) with 38 μm volumetric accuracy and 27 μm point repeatability. The resulting point clouds were processed in MATLAB, where the z-coordinate of each point represents its offset from the reference plane (flat table surface). The deformation plots were generated using MATLAB’s pcshow command. EXPERIMENTAL RESULTS (DEFORMATION MEASUREMENTS) [0086] Experimental validation encompassed all cases evaluated in simulations, i.e., two scan patterns (vector and island) under three boundary conditions, yielding six distinct scenarios. Each scenario compared SmartScan 1.0 with thermal modeling and 2.0 with thermomechanical modeling, resulting in twelve total test plates. Measured deformation profiles were taken for all configurations. By observing the maximum and mean deformation of the plates, one can see that SmartScan 2.0 with thermomechanical modeling yielded a lower maximum deformation (by up to 23.7%) than the corresponding SmartScan 1.0 with thermal modeling scenario for all the cases. This supports the findings from the simulations that showed that SmartScan 2.0 with thermomechanical modeling always yielded a lower value of Jf than SmartScan 1.0 with thermal modeling. We, however, note that with regard to mean deformation, SmartScan 2.0 with thermomechanical modeling generally outperformed or produced nearly the same level of deformation as SmartScan 1.0 with thermal modeling, with the exception of the island scan pattern for Case Study 1 where SmartScan 2.0 with thermomechanical modeling showed a 30.8% higher mean value of deformation than SmartScan 1.0 with thermal modeling. We believe that this has to do with the fact that the thermomechanical objective function (Jf) is focused on minimizing local temperature
Attorney Docket No.2115-008409-WO-POA gradients weighted by local compliance information. Therefore, it may miss global deformation patterns that contribute to higher mean values of deformation. [0087] The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. As used herein, the singular forms "a,” "an," and "the" may be intended to include the plural forms as well, unless the context clearly indicates otherwise. The terms "comprises," "comprising," “including,” and “having,” are inclusive and therefore specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. The method steps, processes, and operations described herein are not to be construed as necessarily requiring their performance in the particular order discussed or illustrated, unless specifically identified as an order of performance. It is also to be understood that additional or alternative steps may be employed. [0088] When an element or layer is referred to as being "on," “engaged to,” "connected to," or "coupled to" another element or layer, it may be directly on, engaged, connected or coupled to the other element or layer, or intervening elements or layers may be present. In contrast, when an element is referred to as being "directly on," “directly engaged to,” "directly connected to," or "directly coupled to" another element or layer, there may be no intervening elements or layers present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between,” “adjacent” versus “directly adjacent,” etc.). As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items. [0089] Although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers and/or sections, these elements, components, regions, layers and/or sections should not be limited by these terms. These terms may be only used to distinguish one element, component, region, layer or section from another region, layer or section. Terms such as “first,” “second,” and other numerical terms when used herein do not imply a sequence or order unless clearly indicated by the context. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the example embodiments.
Attorney Docket No.2115-008409-WO-POA [0090] In the figures, the direction of an arrow, as indicated by the arrowhead, generally demonstrates the flow of information (such as data or instructions) that is of interest to the illustration. For example, when element A and element B exchange a variety of information but information transmitted from element A to element B is relevant to the illustration, the arrow may point from element A to element B. This unidirectional arrow does not imply that no other information is transmitted from element B to element A. Further, for information sent from element A to element B, element B may send requests for, or receipt acknowledgements of, the information to element A. [0091] In this application, including the definitions below, the term “module” or the term “controller” may be replaced with the term “circuit.” The term “module” may refer to, be part of, or include: an Application Specific Integrated Circuit (ASIC); a digital, analog, or mixed analog/digital discrete circuit; a digital, analog, or mixed analog/digital integrated circuit; a combinational logic circuit; a field programmable gate array (FPGA); a processor circuit (shared, dedicated, or group) that executes code; a memory circuit (shared, dedicated, or group) that stores code executed by the processor circuit; other suitable hardware components that provide the described functionality; or a combination of some or all of the above, such as in a system-on-chip. [0092] The module may include one or more interface circuits. In some examples, the interface circuits may include wired or wireless interfaces that are connected to a local area network (LAN), the Internet, a wide area network (WAN), or combinations thereof. The functionality of any given module of the present disclosure may be distributed among multiple modules that are connected via interface circuits. For example, multiple modules may allow load balancing. In a further example, a server (also known as remote, or cloud) module may accomplish some functionality on behalf of a client module. [0093] The term code, as used above, may include software, firmware, and/or microcode, and may refer to programs, routines, functions, classes, data structures, and/or objects. The term shared processor circuit encompasses a single processor circuit that executes some or all code from multiple modules. The term group processor circuit encompasses a processor circuit that, in combination with additional processor circuits, executes some or all code from one or more modules. References to multiple processor circuits encompass multiple processor circuits on discrete dies, multiple processor circuits on a single die, multiple cores of a single processor circuit, multiple
Attorney Docket No.2115-008409-WO-POA threads of a single processor circuit, or a combination of the above. The term shared memory circuit encompasses a single memory circuit that stores some or all code from multiple modules. The term group memory circuit encompasses a memory circuit that, in combination with additional memories, stores some or all code from one or more modules. [0094] The term memory circuit is a subset of the term computer-readable medium. The term computer-readable medium, as used herein, does not encompass transitory electrical or electromagnetic signals propagating through a medium (such as on a carrier wave); the term computer-readable medium may therefore be considered tangible and non-transitory. Non-limiting examples of a non-transitory, tangible computer-readable medium are nonvolatile memory circuits (such as a flash memory circuit, an erasable programmable read-only memory circuit, or a mask read-only memory circuit), volatile memory circuits (such as a static random access memory circuit or a dynamic random access memory circuit), magnetic storage media (such as an analog or digital magnetic tape or a hard disk drive), and optical storage media (such as a CD, a DVD, or a Blu-ray Disc). [0095] The apparatuses and methods described in this application may be partially or fully implemented by a special purpose computer created by configuring a general purpose computer to execute one or more particular functions embodied in computer programs. The functional blocks, flowchart components, and other elements described above serve as software specifications, which can be translated into the computer programs by the routine work of a skilled technician or programmer. [0096] The computer programs include processor-executable instructions that are stored on at least one non-transitory, tangible computer-readable medium. The computer programs may also include or rely on stored data. The computer programs may encompass a basic input/output system (BIOS) that interacts with hardware of the special purpose computer, device drivers that interact with particular devices of the special purpose computer, one or more operating systems, user applications, background services, background applications, etc. [0097] The computer programs may include: (i) descriptive text to be parsed, such as HTML (hypertext markup language), XML (extensible markup language), or JSON (JavaScript Object Notation) (ii) assembly code, (iii) object code generated from source code by a compiler, (iv) source code for execution by an interpreter, (v) source code for compilation and execution by a just-in-time compiler, etc. As examples only, source
Attorney Docket No.2115-008409-WO-POA code may be written using syntax from languages including C, C++, C#, Objective-C, Swift, Haskell, Go, SQL, R, Lisp, Java®, Fortran, Perl, Pascal, Curl, OCaml, Javascript®, HTML5 (Hypertext Markup Language 5th revision), Ada, ASP (Active Server Pages), PHP (PHP: Hypertext Preprocessor), Scala, Eiffel, Smalltalk, Erlang, Ruby, Flash®, Visual Basic®, Lua, MATLAB, SIMULINK, and Python®. [0098] The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.