SmartScan 2.0: An Intelligent Scanning Approach for Reduced Residual Stress and Deformation in LPBF Using a Coupled Linear Thermoelastic Model

1. Introduction

Laser powder bed fusion (LPBF) is a core metal additive manufacturing process used across aerospace, automotive, biomedical, and other applications. It uses a high-intensity laser to selectively fuse layers of metal powder to build 3D geometries [1,2]. Despite its ability to form complex parts with high precision and density, LPBF often produces components with residual stress, distortion, cracks, and related defects. These issues arise from the severe thermal environment inherent to the process, including nonuniform heat input, localized heat accumulation, steep temperature gradients, and rapid cooling, which together drive complex thermo-mechanical interactions [3,4,5,6,7,8]. Post-build heat treatments can mitigate residual stress. However, they add cost and time and do not fully remove as-built defects such as distortion and cracking [8,9,10,11]. Consequently, controlling temperature and stress during fabrication, and reducing residual stress and distortion as they form, is critical to ensuring final part quality [12,13].

Scan strategy is central to managing thermal conditions and limiting defects in LPBF. Many studies optimize scan-related parameters such as laser power [14,15,16,17,18], scan speed [18,19,20], layer thickness [21], and scan pattern [7,18,22,23,24]. These efforts have advanced local process control. Scan sequence is another important aspect of scan strategy. It specifies the order in which the laser visits repeating geometric units (features) defined by a chosen scan pattern. Examples include the order of line segments within parallel hatch lines, the order of islands in a chessboard pattern (see Figure 1), and similar subregions. By reshaping the temporal progression of heat input, scan sequence strongly affects thermal history [25], residual stress [26], and part distortion [27] in LPBF components.

Most existing methods for generating scan sequences are heuristic or empirical. They typically rely on simple rules or optimization schemes that approximate heat transfer using geometric relations among features (e.g., the distance between features) [25,26,28,29,30,31]. Common examples include random island, successive island, and successive chessboard sequences. Mugwagwa et al. [26] showed that the successive chessboard sequence reduced average residual stress by up to 40% relative to a random island sequence. Kruth et al. [25] developed a geometry-driven scan sequence called least heat influence (LHI) that chose the next island farthest from previously scanned regions to limit local heat buildup. Later work by Qin et al. [31] refined island ordering with the goal of improving heat uniformity across the layer. These and other similar approaches show measurable benefits in reducing residual stress and, in some cases, distortion compared to less sophisticated scan sequences. Despite these gains, geometry-based methods remain limited because they are blind to the underlying thermomechanical dynamics of LPBF, relying on static geometric proxies instead [25,28,31]. As a result, they are difficult to generalize across parts, materials, and scan patterns, frequently requiring case-specific tuning or retraining, e.g., [25,26,28,29].

These limitations motivated the introduction of the SmartScan framework which aims to generate scan sequences using physics-based (or data-driven) models and control-theoretic optimization [32,33,34,35,36,37,38,39]. The first attempts at SmartScan (herein called SmartScan 1.0) used a physics-based thermal model to optimize temperature uniformity in LPBF [32,33,34,35,36,37,38]. In several studies, SmartScan 1.0 was shown to outperform state-of-the-art heuristic patterns by reducing temperature gradients, residual stress, and/or deformation. For example, in [37], it decreased temperature inhomogeneity by up to 92%, residual stress by 86%, maximum deformation by 24%, and geometric inaccuracies by 50% relative to state-of-the-art heuristic sequences when tested on 3D geometries. It also maintained computational efficiency compatible with real-time use, and it has been implemented in industrial practice as a plug-in to a commercial slicing software package, enabling successful fabrication of a complex 3D part while preventing catastrophic failure [37]. SmartScan 1.0 has inspired other thermal-model-based optimization approaches, like Reinforced Scan [40], which uses a thermal model coupled with reinforcement learning to optimize scan sequences. With sufficient training, Reinforced Scan can explore a wider search space than SmartScan 1.0 and thus potentially provide better solutions as shown by Qin et al. in [40]. However, the training step in Reinforced Scan is computationally very expensive and time consuming, and it requires retraining for each new geometry or set of process parameters. Both of these limit its practicality and versatility.

Regardless of the approach, a major limitation of thermal-based scan sequence optimization solutions, such as SmartScan 1.0 and Reinforced Scan, is that they neglect the mechanical component of the physics that drives residual stress and deformation in LPBF [40,41,42,43,44]. For example, these methods produce the same optimal scan sequence regardless of the mechanical boundary conditions. To address this limitation, the authors introduced the SmartScan 2.0 framework [39,45]. As illustrated in Figure 2, the overarching goal of SmartScan 2.0 is to employ a linear thermomechanical model and objective function to compute optimal scan sequences for each LPBF layer through computationally efficient, control-theoretic optimization.

A preliminary implementation of this framework - here referred to as SmartScan 2.0 (Pre) - adopted a decoupled linear thermomechanical formulation in which the thermal and mechanical fields were treated separately. Its objective was to minimize local temperature gradients (i.e., in the vicinity of the laser) weighted by local compliance information obtained from a linear finite element mechanical model. This thermomechanical objective was optimized using a computationally efficient control-theoretic procedure similar to that employed in SmartScan 1.0 [39]. Using laser marking experiments on 2D plate geometries, SmartScan 2.0 (Pre) was shown to be responsive to mechanical physics (e.g., boundary conditions) and achieved significant reductions in part distortion relative to SmartScan 1.0 in most cases. However, in some scenarios SmartScan 1.0 outperformed SmartScan 2.0 (Pre); for example, it achieved lower mean deformation in a nontrivial fraction of cases [39], revealing the limitations of the decoupled formulation and the locally defined optimization objective.

To overcome these limitations, we propose an improved and generalized approach that better fulfills the overarching goal of the SmartScan 2.0 framework. For simplicity, we refer to this new approach as SmartScan 2.0 distinguishing it from the preliminary version SmartScan 2.0 (Pre). Specifically, this paper makes the following original contributions:

• It introduces a novel control-theoretic optimization approach for scan sequence generation that minimizes thermally induced elastic deformation across the entire part, layer by layer, based on a sequentially coupled linear thermoelastic model.
• It incorporates a nondimensionalized formulation into the proposed SmartScan 2.0 approach to achieve high computational efficiency and scalability.
• Through 2D metal plate laser marking and 3D cantilever beam printing experimental case studies, it demonstrates substantial reductions in residual stress and deformation using the proposed SmartScan 2.0 compared with SmartScan 1.0 and SmartScan 2.0 (Pre), without sacrificing computational efficiency.

The remainder of this paper is organized as follows. Section 2 presents the sequentially coupled linear thermoelastic model, objective function, and the proposed control-theoretic optimization approach. Section 3 then provides a brief theoretical comparison of SmartScan 2.0 with SmartScan 1.0 and SmartScan 2.0 (Pre), ensuring that their similarities and differences are clear. Section 4 introduces the nondimensionalization strategy, grounded in similarity theory, that enables the computational efficiency and scalability of SmartScan 2.0. Section 5 reports experimental validation results benchmarking SmartScan 2.0 against SmartScan 1.0 and SmartScan 2.0 (Pre). Section 6 concludes the paper and outlines future research directions.

2. Proposed SmartScan 2.0

2.1. Overview

The goal of SmartScan 2.0 is to develop an effective and computationally efficient thermomechanical proxy for solving the inverse problem in LPBF; i.e., determining the optimal process inputs (here, the scan sequence) that yield desired outputs such as minimal residual stress and distortion. A proxy is necessary because this inverse problem is far more challenging than the forward problem of predicting outputs from known inputs. Although high-fidelity thermomechanical models capable of accurately predicting residual stress and distortion exist [46,47,48,49], they are too computationally intensive to be practical for inverse optimization, particularly in scan sequence optimization, where the number of possible sequences grows factorially with the number of features in a layer.

To enable tractable optimization, SmartScan 2.0 adopts a linear thermomechanical formulation and therefore does not explicitly model nonlinearities such as strong temperature dependence of material properties, plasticity, or creep. The resulting linear thermoelastic model serves as a simplified but physics-based representation of the process. As a proxy for reducing residual stress and distortion using the simplified model, the optimization targets the minimization of thermally induced elastic deformation during printing. The underlying rationale is that residual stress and distortion arise from plastic deformation, which occurs when thermally induced elastic deformation generates stresses that exceed the yield strength. Therefore, minimizing elastic deformation increases the likelihood of reducing the regions and extent where yielding occurs and, in turn, reducing the magnitude of residual stress and distortion. While this proxy is approximate, as we will show, it provides an effective and computationally efficient alternative to the thermal-only proxies used in prior work. The remainder of this section details the linear thermoelastic model, the resulting objective function, and how these enable an elegant control-theoretic solution for real-time scan sequence optimization.

2.2. Thermal Model

The thermal model is adopted from our prior work on SmartScan 1.0 [37,39]. The model accounts for conductive and convective heat transfer within the part and between the part and its surroundings, while incorporating several simplifying assumptions. In particular, we neglect radiative heat transfer, latent heat effects, Marangoni convection, and other melt pool phenomena [50,51]. We also assume temperature-independent material parameters, yielding a linear model that facilitates efficient optimization. Under these assumptions, heat transfer within the printed part, characterized by conductivity $k_t$ and diffusivity $\alpha$, is governed by the equation:

$${\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}+{\displaystyle \frac{u}{k_t}}={\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial T}{\partial t}}$$

(1)

where T represents temperature, $(x,y,z)$ are the spatial coordinates, t is time, and u denotes the power per unit volume. The equation can be discretized using the finite difference method (FDM) with discrete grid sizes $\Delta x$, $\Delta y$, $\Delta z$ and an explicit time step $\Delta t$ to yield:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{T(i+1,j,k,l)+T(i-1,j,k,l)-2T(i,j,k,l)}{\Delta x^2}}\hfill \\ \hfill & +{\displaystyle \frac{T(i,j+1,k,l)+T(i,j-1,k,l)-2T(i,j,k,l)}{\Delta y^2}}\hfill \\ \hfill & +{\displaystyle \frac{T(i,j,k+1,l)+T(i,j,k-1,l)-2T(i,j,k,l)}{\Delta z^2}}\hfill \\ \hfill & +{\displaystyle \frac{u(i,j,k,l)}{k_t}}={\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{T(i,j,k,l+1)-T(i,j,k,l)}{\Delta t}}\hfill \end{array}$$

(2)

where $i,j,k$ are spatial indices for the elements, and l represents the temporal index. The temperature at position $i,j,k$ and time step l is denoted as $T(i,j,k,l)$. The simplified FDM thermal model, which includes convection boundary conditions on the upper surface of the part (details in Ref. [37]), can be written as a state equation:

$$\mathbf{T}(l+1)=\mathbf{A}\mathbf{T}\left(l\right)+\mathbf{Bu}\left(l\right)$$

(3)

where $\mathbf{T}\left(l\right)\in {\mathbb{R}}^{(n_e+1)\times 1}$ is the temperature vector containing the temperatures of all $n_e$ elements and the constant ambient temperature $T_a$ at time step l, $\mathbf{A}\in {\mathbb{R}}^{(n_e+1)\times (n_e+1)}$ is the state matrix, $\mathbf{B}\in {\mathbb{R}}^{(n_e+1)\times n_e}$ is the input matrix, and $\mathbf{u}\left(l\right)\in {\mathbb{R}}^{n_e\times 1}$ represents the power input vector with non-zero values only at the elements actively heated by the laser at time step l. The model incorporates convection to the ambient temperature $T_a$, with coefficient h, on its top surfaces. Additionally, the substrate is treated as a heat sink at a constant temperature of $T_b$. The side edges are considered to be insulated, as the thermal conductivity of the metal powder is extremely low and can be neglected. More details about the setup and validation of this model can be found in [37].

Figure 3. Simplified thermal finite difference model (FDM) incorporating the assumptions used for SmartScan 2.0, as reported in Ref. [37].

Figure 3. Simplified thermal finite difference model (FDM) incorporating the assumptions used for SmartScan 2.0, as reported in Ref. [37].

Recall that the features refer to the repeated geometric units in the scan pattern (e.g., vectors or islands). For a total of $n_F$ features, let each feature f require $N_f$ time steps for complete scanning, with its time steps recorded in a set $g_f=\{l_{f,0},l_{f,1},\dots ,l_{f,N_f}\}$. When scanning each feature $f\in \{1,2,\dots ,n_F\}$, the temperature at any time point $l_{f,m}$ where $m\in \{0,1,\dots ,N_f\}$ can be expressed as:

$$\begin{array}{cc}\hfill & \mathbf{T}\left(l_{f,m}\right)={\mathbf{A}}_{l_{f,m}}\mathbf{T}\left(l_{f,0}\right)+{\mathbf{b}}_{l_{f,m}}\hfill \\ \hfill & {\mathbf{A}}_{l_{f,m}}={\mathbf{A}}^m;{\mathbf{b}}_{l_{f,m}}={\Sigma }_{n=0}^{m-1}{\mathbf{A}}^{m-1-n}\mathbf{B}\mathbf{u}\left(l_{f,n}\right)\hfill \end{array}$$

(4)

where $l_{f,0}$ denotes the start time of scanning feature f, and m is the number of time steps that have elapsed at the point of interest. Note that the thermal model assumes zero transition time between consecutive scanned features for simplicity. This assumption is reasonable, as the jump speed is typically much larger than the scanning speed.

2.3. Mechanical Model

In this section, we provide a comprehensive mathematical derivation for addressing pure linear elastic deformation caused by a thermal field, employing the finite element method (FEM). We consider a quasi-static problem where deformation results exclusively from temperature variations, as derived from the calculations in Section 2.2, without any external mechanical loads. The temperature field is represented as a nodal vector $\mathbf{T}\left(l_{f,m}\right)$, and our objective is to determine the corresponding displacement field $\mathbf{W}\left(l_{f,m}\right)$.

The linear thermoelastic problem is described by the equilibrium equation [52], assuming no body forces are present:

$$\nabla ·\sigma =\mathbf{0}$$

(5)

where $\sigma$ represents the stress tensor. For isotropic linear elastic materials that account for thermal effects, the constitutive relationship is given by:

$$\sigma =\mathbf{D}(\epsilon -{\epsilon }_{th})$$

(6)

where $\mathbf{D}$ is the elasticity matrix, which connects stress to strain for the material, $\epsilon$ is the total strain tensor, and ${\epsilon }_{th}=\beta \Delta T\mathbf{I}$ is the thermal strain tensor. In this expression, $\beta$ denotes the coefficient of thermal expansion, $\Delta T=T-T_{ref}$ denotes the temperature change relative to the reference temperature $T_{ref}$, and $\mathbf{I}$ denotes the identity tensor.

The kinematic relationship, which relates the strain tensor to the displacement field $\mathbf{w}$, can be expressed as:

$$\epsilon =\frac{1}{2}\left(\nabla \mathbf{w}+{(\nabla \mathbf{w})}^{\mathsf{T}}\right)$$

(7)

where $\nabla \mathbf{w}$ is the displacement gradient tensor.

In FEM, the three-dimensional domain $\mathsf{\Omega }\subset {\mathbb{R}}^3$ is discretized into a mesh of finite elements. For simplicity, this study uses 8-node hexahedral elements [53] that align with the grid described in Section 2.2. The principle of virtual work is applied to derive the weak form of the thermoelastic problem [54], expressed as:

$${\int }_{\mathsf{\Omega }}\delta {\epsilon }^{\mathsf{T}}\mathbf{D}(\epsilon -{\epsilon }_{th})dV=0$$

(8)

where $\delta \epsilon$ represents the virtual strain field and $dV$ denotes the differential volume element. This formulation assumes no external mechanical loads are applied. By discretizing the domain and assembling contributions from all elements, the global system of equations is obtained as:

$$\mathbf{K}\mathbf{W}={\mathbf{F}}_{th}$$

(9)

with the global stiffness matrix $\mathbf{K}={\sum }_e{\mathbf{K}}^e$, where the element stiffness matrix is ${\mathbf{K}}^e={\int }_{{\mathsf{\Omega }}^e}{\left({\mathbf{B}}^e\right)}^{\mathsf{T}}{\mathbf{D}}^e{\mathbf{B}}^{e}dV$, and the global thermal load vector ${\mathbf{F}}_{th}={\sum }_e{\mathbf{F}}_{th}^e$, where the element thermal load vector is ${\mathbf{F}}_{th}^e={\int }_{{\mathsf{\Omega }}^e}{\left({\mathbf{B}}^e\right)}^{\mathsf{T}}{\mathbf{D}}^e{\epsilon }_{th}^{e}dV$. Additionally, ${\mathbf{B}}^e=\nabla {\mathbf{N}}^e$ is the finite-element strain–displacement matrix derived from the element shape functions. This study employs the classic trilinear shape functions for 8-node hexahedral elements [53]. For isotropic materials in three-dimensional space, the thermal strain is defined as ${\epsilon }_{th}^e=\beta \Delta T^e\mathbf{m}$ , where $\Delta T^e$ denotes the temperature change in element e relative to the reference temperature $T_{ref}$, and $\mathbf{m}={[1,1,1,0,0,0]}^{\mathsf{T}}$ is the vector accounting for thermal strain components.

With the global nodal temperature change $\Delta \mathbf{T}=\mathbf{T}-{\mathbf{T}}_0$, where ${\mathbf{T}}_0$, taken as $T_{ref}$, denotes the initial nodal temperature at the start of scanning for each layer, and assuming that ${\mathbf{F}}_{th}$ is zero at the initial point, we define the thermal load matrix $\mathbf{L}$ such that ${\mathbf{F}}_{th}=\mathbf{L}\Delta \mathbf{T}$. At the element level, ${\mathbf{F}}_{th}^e$ can be expressed as:

$${\mathbf{F}}_{th}^e={\int }_{{\mathsf{\Omega }}^e}{\left({\mathbf{B}}^e\right)}^{\mathsf{T}}{\mathbf{D}}^e{\epsilon }_{th}^{e}dV={\int }_{{\mathsf{\Omega }}^e}{\left({\mathbf{B}}^e\right)}^{\mathsf{T}}{\mathbf{D}}^e\beta \Delta T^e\mathbf{m}dV={\mathbf{L}}^e\Delta T^e$$

(10)

Upon global assembly, $\mathbf{L}={\sum }_e{\mathbf{L}}^e$. At time point $l_{f,m}$, ${\mathbf{F}}_{th}\left(l_{f,m}\right)$ can be represented as

$${\mathbf{F}}_{th}\left(l_{f,m}\right)=\mathbf{L}\Delta \mathbf{T}\left(l_{f,m}\right)=\mathbf{L}(\mathbf{T}\left(l_{f,m}\right)-{\mathbf{T}}_0)$$

(11)

Assuming that $\mathbf{K}$ is invertible (as ensured by boundary conditions, such as fixed displacements on portions of the boundary $\partial \mathsf{\Omega }$), $\mathbf{W}\left(l_{f,m}\right)$ can be solved through

$$\mathbf{W}\left(l_{f,m}\right)={\mathbf{K}}^{-1}{\mathbf{F}}_{th}\left(l_{f,m}\right)={\mathbf{K}}^{-1}\mathbf{L}(\mathbf{T}\left(l_{f,m}\right)-\mathbf{T}\left(l_{f,0}\right))={\mathbf{K}}^{-1}\mathbf{L}\mathbf{T}\left(l_{f,m}\right)-{\mathbf{W}}_0$$

(12)

where ${\mathbf{W}}_0={\mathbf{K}}^{-1}\mathbf{L}\mathbf{T}\left(l_{f,0}\right)$. Boundary conditions are applied by modifying $\mathbf{K}$, for example, through row and column elimination or penalty methods [55].

2.4. Objective Function and Optimization

SmartScan 2.0 identifies the optimal scanning sequence, ensuring that each feature is scanned exactly once. The objective is to minimize the ${\ell }_2$ norm of the elastic deformation vector $\mathbf{W}\left(l_{f,N_f}\right)$, which represents the deformation field of the entire part just after the scanning of feature f within each layer is completed, as calculated in Equation (12). Since ${\mathbf{W}}_0$ is a fixed constant, it can be omitted from the objective function to simplify the optimization. Accordingly, the objective function for SmartScan 2.0 (called the elastic deformation metric) can be represented as:

$$D\left(l_{f,N_f}\right)={∥\mathbf{W}\left(l_{f,N_f}\right)+{\mathbf{W}}_0∥}_2={∥{\mathbf{K}}^{-1}\mathbf{L}\mathbf{T}\left(l_{f,N_f}\right)∥}_2$$

(13)

Based on the feature-level state-space representation provided in Equation (4) and the elastic deformation metric defined in Equation (12), the optimization problem applied to the layer of interest can be formulated as follows:

$$\begin{array}{ccc}\hfill & f^{*}=arg\underset{f}{min}D\left(l_{f,N_f}\right)\hfill & \\ \hfill & \mathrm{s}.\mathrm{t}.\mathbf{T}\left(l_{f,N_f}\right)={\mathbf{A}}_{l_{f,N_f}}\mathbf{T}\left(l_{f,0}\right)+{\mathbf{b}}_{l_{f,N_f}}\hfill \end{array}$$

(14)

The optimization problem presented in Equation(14) can be reformulated as follows:

$$\begin{array}{cc}\hfill & {∥{\mathbf{K}}^{-1}\mathbf{L}\mathbf{T}\left(l_{f,N_f}\right)∥}_2^2\hfill \\ \hfill & ={∥{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{A}}_{l_{f,N_f}}\mathbf{T}\left(l_{f,0}\right)+{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{b}}_{l_{f,N_f}}∥}_2^2\hfill \\ \hfill & =\mathbf{T}{\left(l_{f,0}\right)}^{\mathsf{T}}{\mathbf{A}}_{l_{f,N_f}}^{\mathsf{T}}{\mathbf{L}}^{\mathsf{T}}{\left({\mathbf{K}}^{-1}\right)}^{\mathsf{T}}{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{A}}_{l_{f,N_f}}\mathbf{T}\left(l_{f,0}\right)\hfill \\ \hfill & +2\mathbf{T}{\left(l_{f,0}\right)}^{\mathsf{T}}{\mathbf{A}}_{l_{f,N_f}}^{\mathsf{T}}{\mathbf{L}}^{\mathsf{T}}{\left({\mathbf{K}}^{-1}\right)}^{\mathsf{T}}{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{b}}_{l_{f,N_f}}\hfill \\ \hfill & +{\mathbf{b}}_{l_{f,N_f}}^{\mathsf{T}}{\mathbf{L}}^{\mathsf{T}}{\left({\mathbf{K}}^{-1}\right)}^{\mathsf{T}}{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{b}}_{l_{f,N_f}}\hfill \end{array}$$

(15)

Accordingly, the problem is expressed as:

$$\begin{array}{ccc}\hfill & f^{*}=arg\underset{f}{min}{\lambda }_f\hfill & \\ \hfill & \mathrm{s}.\mathrm{t}.{\lambda }_f={ϵ}_f+{\gamma }_f\mathbf{T}\left(l_{f,0}\right)+{\mathbf{T}}^{\mathsf{T}}\left(l_{f,0}\right){\delta }_f\mathbf{T}\left(l_{f,0}\right)\hfill \\ \hfill & {ϵ}_f\triangleq {\mathbf{b}}_{l_{f,N_f}}^{\mathsf{T}}{\mathbf{L}}^{\mathsf{T}}{\left({\mathbf{K}}^{-1}\right)}^{\mathsf{T}}{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{b}}_{l_{f,N_f}}\hfill \\ \hfill & {\gamma }_f\triangleq 2{\mathbf{b}}_{l_{f,N_f}}^{\mathsf{T}}{\mathbf{L}}^{\mathsf{T}}{\left({\mathbf{K}}^{-1}\right)}^{\mathsf{T}}{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{A}}_{l_{f,N_f}}\hfill \\ \hfill & {\delta }_f\triangleq {\mathbf{A}}_{l_{f,N_f}}^{\mathsf{T}}{\mathbf{L}}^{\mathsf{T}}{\left({\mathbf{K}}^{-1}\right)}^{\mathsf{T}}{\mathbf{K}}^{-1}\mathbf{L}{\mathbf{A}}_{l_{f,N_f}}\hfill \end{array}$$

(16)

where ${ϵ}_f$, ${\gamma }_f$, and ${\delta }_f$ can be precomputed for each feature $f\in 1,2,\dots ,n_F$ in a layer. The feature $f^{*}$ that minimizes the elastic deformation metric can thus be calculated as the index corresponding to the smallest element in the vector ${\lambda }_f$. Figure 4 shows the SmartScan 2.0 optimization workflow. Note that key parameters listed at the top of the flowchart, e.g., the stiffness matrix K, need to be updated after every layer of the optimization, as the part is built.