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WO2023076636A1 - Modélisation thermique de fabrication d'additif - Google Patents

Modélisation thermique de fabrication d'additif Download PDF

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Publication number
WO2023076636A1
WO2023076636A1 PCT/US2022/048296 US2022048296W WO2023076636A1 WO 2023076636 A1 WO2023076636 A1 WO 2023076636A1 US 2022048296 W US2022048296 W US 2022048296W WO 2023076636 A1 WO2023076636 A1 WO 2023076636A1
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Prior art keywords
nodes
node
representation
heat
temperature
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Kevin Cole
Alex RIENSCHE
Prahalada Rao
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NuTech Ventures Inc
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NuTech Ventures Inc
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B33—ADDITIVE MANUFACTURING TECHNOLOGY
    • B33Y—ADDITIVE MANUFACTURING, i.e. MANUFACTURING OF THREE-DIMENSIONAL [3-D] OBJECTS BY ADDITIVE DEPOSITION, ADDITIVE AGGLOMERATION OR ADDITIVE LAYERING, e.g. BY 3-D PRINTING, STEREOLITHOGRAPHY OR SELECTIVE LASER SINTERING
    • B33Y50/00—Data acquisition or data processing for additive manufacturing
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B22—CASTING; POWDER METALLURGY
    • B22F—WORKING METALLIC POWDER; MANUFACTURE OF ARTICLES FROM METALLIC POWDER; MAKING METALLIC POWDER; APPARATUS OR DEVICES SPECIALLY ADAPTED FOR METALLIC POWDER
    • B22F10/00—Additive manufacturing of workpieces or articles from metallic powder
    • B22F10/80—Data acquisition or data processing
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B22—CASTING; POWDER METALLURGY
    • B22F—WORKING METALLIC POWDER; MANUFACTURE OF ARTICLES FROM METALLIC POWDER; MAKING METALLIC POWDER; APPARATUS OR DEVICES SPECIALLY ADAPTED FOR METALLIC POWDER
    • B22F10/00—Additive manufacturing of workpieces or articles from metallic powder
    • B22F10/20—Direct sintering or melting
    • B22F10/25—Direct deposition of metal particles, e.g. direct metal deposition [DMD] or laser engineered net shaping [LENS]
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B22—CASTING; POWDER METALLURGY
    • B22F—WORKING METALLIC POWDER; MANUFACTURE OF ARTICLES FROM METALLIC POWDER; MAKING METALLIC POWDER; APPARATUS OR DEVICES SPECIALLY ADAPTED FOR METALLIC POWDER
    • B22F10/00—Additive manufacturing of workpieces or articles from metallic powder
    • B22F10/20—Direct sintering or melting
    • B22F10/28—Powder bed fusion, e.g. selective laser melting [SLM] or electron beam melting [EBM]
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2113/00—Details relating to the application field
    • G06F2113/10—Additive manufacturing, e.g. 3D printing
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/08—Thermal analysis or thermal optimisation

Definitions

  • This disclosure relates to improvements to additive manufacturing processes.
  • Additive manufacturing e.g., three-dimensional printing
  • additive manufacturing is a process in which layers of material are sequentially applied and fused together. Inadequate heat dissipation can lead to failure of AM manufactured parts.
  • AM/3D printing offers unparalleled advantages over conventional manufacturing, including greater design freedom and a lower lead time.
  • AM parts in safety-critical industries, such as aerospace and biomedical, is limited by the tendency of the process to create flaws that can lead to sudden failure during use.
  • the root cause of flaw formation in metal AM parts is linked to the temperature inside the part during the process, called the thermal history.
  • the thermal history is a function of the process parameters and part design.
  • FIG. 1 A illustrates an LPBF process schematic
  • FIG. IB illustrates a DED process schematic in which metal powder is sprayed via nozzles and fused onto a substrate by a laser beam.
  • FIG. 2 illustrates lack-of-fusion defects in a titanium part made with LPBF. These same defects are common in DED.
  • FIG. 3 illustrates salient thermal phenomena in DED include conductive, convective, and radiative heat transfer.
  • FIG. 4 illustrates experimental data for each deposition case along with the corresponding thermocouple location and dwell time.
  • FIG. 5 A illustrates a schematic of the clamped substrate in relation to the thin wall
  • FIG. 5B illustrates the wall constructed in case A and case B
  • FIG. 5C illustrates the wall constructed in case C.
  • FIG. 6 illustrates the thermocouple locations for each case.
  • FIG. 7A illustrates an example process for modeling heat flux in AM parts that can be executed in accordance with implementations of the present disclosure.
  • FIG. 7B illustrates representation of the four steps in the graph-theoretic approach for the DED process.
  • FIG. 7C illustrates a graphical illustration of the conversion of a three- dimensional model of an AM part into a node representation.
  • FIG. 7D illustrates a graphical representation of nodes being sorted into layers.
  • FIG. 7E illustrates a graphical representation of nodes being sorted into hatches.
  • FIG. 7F illustrates a graphical representation of generating a weight for a node of an adjacency matrix based on neighbouring nodes.
  • FIG. 8 illustrates a block-by-block heating scheme used in graph theory DED simulation.
  • FIG. 9 illustrates snapshots of the graph theory-based simulation for each case.
  • the lack of dwell time in Case B and Case C leads to accumulation of heat in the top layers of the part.
  • FIG. 10 illustrates an example boundary node with thermal resistance for heat loss to surroundings at T m .
  • FIG. 11 illustrates diagrams of an example geometry for heat flow between nodes for (a) 1-D uniform grid (b) 3-D uniform rectangular grid (c) 3-D random grid.
  • FIG. 12 illustrates a schematic diagram of a slab with piecewise internal heating, one insulated boundary, and one convection boundary used in a first example simulation case.
  • FIG. 18 illustrates a schematic diagram of a geometry for a third simulation example case of a parallelepiped with piecewise initial condition and convection heat loss at the boundaries.
  • FIG. 19 illustrates a diagram showing locations for observing temperature in the parallelepiped of the third Example simulation case.
  • FIG. 21 illustrates a graph of average gain versus node density (n/vol) at several Biot numbers. For each node density n/vol, ten block-random grids were used to find averages; error bars show the variance.
  • FIG. 23 illustrates a schematic diagram of open architecture LPBF platform and the LWIR camera setup used to perform a comparison of the SG simulation method with a test build of a laser powder bed fusion (LPBF) process.
  • LPBF laser powder bed fusion
  • FIG. 24 illustrates a diagram of the inverted half cone geometry used to experimentally validate the SG simulation method disclosed herein and a photograph of a build plate of two post build half cone geometry parts.
  • FIG. 25A illustrates an infrared image showing the 9 pixel by 9 pixel area over which the surface temperature trends are averaged for the C45 cone-shaped test artifact.
  • FIG. 25B illustrates a graph of surface temperature trends for the entire duration of the build.
  • FIG. 25 C illustrates a graph of a zoomed in area of the temperature trends over three process cycles.
  • FIG. 25D illustrates a graph showing the end-of-cycle surface temperature for the duration of build corresponding to the C45 cone-shaped test artifact.
  • FIG. 25E illustrates a diagram of process events that cause the three epochs observed in the temperature trends shown in in FIG. 25C
  • FIG. 26 illustrates graphs of the end-of-cycle temperature histories for the inverted half cone parts C40 and C45.
  • FIG. 27 illustrates a graph representing results of a convergence study on node density for a 50-layer simulated build of the C40 inverted half cone, compared to experimental data.
  • FIG. 28 illustrates an example computing system, according to implementations of the present disclosure.
  • AM Metal additive manufacturing
  • LPBF laser powder bed fusion
  • DED directed energy deposition
  • LOF also appears in DED.
  • DED DED
  • the present disclosure provides a graph theory approach for thermal modeling for AM that employs discrete Green’s functions through treatment of a generalized boundary condition.
  • the graph theory approach for thermal modeling in AM has been published in the context of an LPBF thermal modeling process disclosed in International Patent Application PCT/US2019/051192, SIMULATING HEAT FLUX IN ADDITIVE MANUFACTURING, and US Patent Application No. 17/499,402, THERMAL MODELING OF ADDITIVE MANUFACTURING USING GRAPH THEORY, the contents of which are hereby incorporated by reference in their entirety.
  • the spectral graph method the heat equation is solved over a discrete set of nodes.
  • the spectral graph method is combined with discrete Green’s functions through treatment of a generalized boundary condition.
  • FIG. 3 outlines the salient thermal phenomena in DED.
  • the phenomena labeled 3, 4, and 5 are unique to DED and are not present in the LPBF process. Consequently, certain heat transfer-related assumptions made in the context of the LPBF process to aid computation in LPBF simulations should be relaxed for the DED process.
  • a difference between LPBF and DED is that in the former process, the part is surrounded by unfused powder material, viz., an insulating medium. Hence, heat loss on the top surface of the part occurs through radiation and forced convective heat transfer from the melt pool. Heat loss in the rest of the LPBF part occurs largely through conduction, albeit, heat loss through free convection occurs at the part-powder boundaries given air gaps in the unfused metal powder surrounding the part.
  • the part in DED is surrounded not by metal powder but by an inert gas, and therefore heat is lost to the surroundings through convection and radiation from all surfaces.
  • Convection involves both free and forced convection, as the metal powder is delivered to the substrate via an inert carrier gas, such as argon. Consequently, for a comprehensive model of part-level thermal history in DED, an accurate simulation will account for heat loss through conduction, e.g., both free and forced convection, and radiation.
  • the thermal simulation has also been used for wire arc additive manufacturing (WAAM) for which the heat source is an electric arc, not a laser. This is a variation of the DED process as the growing part is surrounded by gas, not powder.
  • WAAM wire arc additive manufacturing
  • the laser heat source-related assumptions in LPBF do not carry over to the DED process because the scan velocity and spot size (beam diameter) of the laser, and layer thickness are considerably different.
  • LPBF the laser is moved by a set of mirrors and the mirrors are moved by galvanometers.
  • DED the laser head is translated by the physical motion of computer numerical controls (CNC), or in other words, CNC-based axes. Consequently, the scan velocity of the laser in DED is ten times slower compared to LPBF - the scan speed of the laser in LPBF is typically 200 to 500 mm- s’ 1 ; in DED, the scan speed is on the order of 10 mm- s’ 1 .
  • CNC computer numerical controls
  • the typical layer thickness is around 50 pm in LPBF, compared to -100 pm to 200 pm for DED.
  • the laser beam diameter in the DED process is typically nearer to the millimeter range compared to -50 pm to 100 pm in LPBF.
  • the higher laser scan velocity and smaller layer thickness of LPBF are advantageous for reducing the computation time.
  • researchers often simulate the deposition of multiple layers at a time in LPBF (called the super-layer or meta-layer assumption) to reduce the computation time.
  • the super-layer or meta-layer assumption uses the meta-layer assumption in an FE-model to predict thermal-induced deformation in LPBF. Meta-layers ranging from 12 to as large as 50 times the actual layer thickness (50 pm) are simulated. Using this technique a model predicts distortion within 5% of measurements, despite simulating the deposition of -15 layers at a time.
  • the slow scan speed and large laser spot size of DED ensure that the melt pool has a large diameter and penetrates deeper into the previous layers compared to LPBF. Consequently, the meta-layer assumption is not viable in DED.
  • the present disclosure provides a graph theory approach for thermal modeling for AM that employs discrete Green’s functions through treatment of a generalized boundary condition.
  • the graph theory approach for thermal modeling in AM has been published in the context of an LPBF thermal modeling process disclosed in International Patent Application PCT/US2019/051192, SIMULATING HEAT FLUX IN ADDITIVE MANUFACTURING, and US Patent Application No. 17/499,402, THERMAL MODELING OF ADDITIVE MANUFACTURING USING GRAPH THEORY, the contents of which are hereby incorporated by reference in their entirety.
  • spectral graph (SG) method lies in the ease with which any geometry may be represented by a collection of nodes dispersed throughout the body.
  • the inventors have shown in research that the SG method may be computed faster than commercial finite element codes for comparable precision, for thermal simulation of additive manufacturing.
  • the research involved insulated boundaries, with boundary heat loss included as an adjustment to the boundary temperatures, external to the SG method.
  • Green’s function (GF) method disclosed herein is that several types of heating conditions may be addressed with straightforward steps if the GF is known. Discrete building- block solutions can be constructed from the GF to treat heating conditions that vary over space and over time.
  • the geometric universality and computational speed of the SG method is combined with the multiple-heating capability and mathematical rigor of the GF method.
  • the improved method can treat boundary conditions of type 1, 2, and 3 (discussed below), under a variety of heat-addition conditions, and has great potential to provide rapid thermal simulations of a variety of industrial processes.
  • FIG. 7A depicts an example process 700 for modeling heat flux in AM parts made by a DED process.
  • Process 700 can be executed in accordance with implementations of the present disclosure.
  • the example process 700 can be implemented, for example, by one or more computing systems.
  • Exemplary computing systems can include, but are not limited to, a super computer, a desktop computer, a laptop computer, or a tablet computer.
  • the example process 700 can be provided by one or more computer-executable programs executed using one or more computing devices.
  • the example process 700, or portions thereof can be provided by one or more programs executed by a computing system.
  • FIG. 7B illustrates graphical representations of the steps of process 700.
  • the computing system converts a three-dimensional model of an AM part into a node representation of the AM part (702). For example, a fixed number of spatial locations within the part are randomly sampled. The heat flux through the part is observed at these fixed spatial locations, termed nodes. In other words, the volume of the part to be simulated is randomly populated with nodes; the number of nodes is set at a certain number per unit volume (called node density). This discretization results in each node having a unique Cartesian (x, y, z) coordinate, i.e., the location of each node is spatially defined within the part. The random sampling of the nodes reduces the computational burden of the approach. The number of nodes is contingent on the geometry of the part.
  • FIG. 7C depicts a graphical illustration of the conversion of a three-dimensional model of an AM part into a node representation.
  • each layer is divided into a hatch (in the case of a thin wall, each layer has only one hatch), and each hatch is further divided into discrete blocks (volumes) with a fixed height and length, with breadth equal to the hatch width.
  • An example block is illustrated in FIG. 8.
  • the computing system generates network graph of the AM part based on the node representation (704).
  • Step 704 can include four sub-steps. The substeps can include: sorting nodes based on z-coordinates to define layers of the object (704a).
  • FIG. 7D depicts a graphical representation of nodes being sorted into layers. Sorting nodes based on y-coordinates to define hatches (704b).
  • FIG. 7E depicts a graphical representation of nodes being sorted into hatches. Sorting nodes based on X- coordinates to define blocks (704c).
  • FIG. 8 depicts a graphical representation of nodes being sorted into blocks.
  • the hatch and block widths and lengths are related to the size of a DED nozzle being simulated. For instance, the width of the hatch and block is comparable to the size of DED layer printed by a particular type or size nozzle.
  • block lengths can be related to both nozzle size and the velocity at which the simulated nozzle is moved (or expected to be moved) across a part. For example, block length can be determined based on the nozzle velocity and a predetermined time step size. In such implementations, the block length would be the product of the nozzle velocity and the size of the time step.
  • the single track or hatch that composes each deposited layer will be broken up into five equal blocks. These discrete blocks are 7.84 mm long, 3 mm wide, and 0.1806 mm thick. There are a total of 2830 blocks in the part (see e.g., FIGS. 7B-7E). The reason for dividing a hatch into blocks is explained in the context of Steps 706-708 and depicted in FIG. 8. Since the nodes are populated in a random manner, there is a degree of uncertainty in the model predictions. This uncertainty can be quantified by repeating the simulations three times for each case.
  • FIG. 7F depicts a graphical representation of generating a weight for a node of an network graph based on neighbouring nodes.
  • the computing system calculates a graph over the set of nodes sampled in Step 702.
  • Each node is connected to its nearest neighboring nodes within a E-radius.
  • E [mm] as describing the radius of a sphere around a node at the center of the sphere. Nodes that fall within the volume of the sphere are connected to the node at the center of the sphere. Nodes that are outside of the sphere are not connected.
  • o is the standard deviation of all the pairwise distances between nodes
  • the exponential term is the Gaussian function that scales the distance between the nodes in the part between 0 and 1. Nodes that are farther away from each other have an edge with a smaller weight connecting each other; nodes closer to each other are connected with an edge with a larger weight,
  • the neighborhood distance E is a heuristic tunable parameter in the graph theory model, that needs to be calibrated for different material types.
  • the calibration procedure for E is described in below.
  • the matrix formed by placing a L j in a row i and column j is called the network graph, which is a positive symmetric matrix. From the network graph, the degree of a node 7T 7 is calculated by summing the corresponding i th row (or column) of A - The graph Laplacian at node is defined as , and the Laplacian matrix is obtained Finally, the eigenspectra of the graph Laplacian matrix (L) will be computed as are the eigenvectors and A are the eigenvalues of L.
  • each layer is simulated block-by-lock by applying simulated heat to a layer of the AM part as represented by the network graph (706).
  • the DED simulation proceeds by heating the nodes in a block, before proceeding to the nodes in the next block.
  • a time step involves heating of nodes inside a block, one block at a time.
  • FIG. 8 demonstrates the block-by -block heating scheme.
  • the laser scan velocity is 8.5 mm-s’ 1 and the length of each block is 7.84 mm long, the time to step between blocks is 0.922 s.
  • the diffusion of heat to other layers of the AM part as represented by the network graph is estimated (708).
  • the heat diffusion is simulated through the network graph that was constructed in step 704.
  • the heat diffusion between nodes is simulated based on the edge weight values connecting node pairs. Conduction is the sole mode of heat transfer between the nodes.
  • the only nodes active during this step are the ones located within layers and blocks that have already been deposited. Other nodes that are in subsequent blocks and layers remain inactive and therefore unable to transfer heat.
  • the deposition of the next block is simulated. This process is repeated for every block and every layer in the part.
  • the mathematical implications of the approach will be summarized here by only including the final derived equation, shown in Eqn. (3).
  • time step t b (e.g., 0.922 s), viz, the time required to process each block, the temperature of each active node is contained in the temperature matrix T b .
  • the time step t b in the simulation is not fixed but can vary depending on the laser speed and chosen block size.
  • the temperature following heat transfer by conduction (T c ) is defined as a function of the Laplacian eigenvectors and eigenvalues (A) of the network graph over active nodes, where To is the melt pool temperature, and g is a tunable parameter called the gain factor (g).
  • the gain factor (g) scales the rate of thermal diffusivity or heat flux between nodes. A higher gain factor increases the rate of thermal diffusion through the part, i.e., the larger the gain factor, the faster the heat will dissipate through the part by conduction. The procedure to calibrate the gain factor is reported in below.
  • the calibration process involves comparison of the simulated temperature found by Eq. (3) with a test solution or test experiment.
  • steps 708 and 710 we described the heating of only those blocks in the topmost layer. However, in DED the block immediately below the block being heated is also at an elevated temperature as the laser penetrates deeper.
  • the temperature of every node is recorded in a vector T b . This is repeated until an entire layer is simulated. After the process reaches the end of the layer, the system simulates heat dissipation by conduction and by convection in steps of, e.g., 1 second, iteratively for a period equal to the dwell time (t d ).
  • t d 20 s for Case A
  • t d 3 s for Case B and Case C.
  • the time t 1 s, and therefore the pair of equations are looped together for 20 times for Case A, and 3 times for Cases B and C to simulate the total dwell time between layers.
  • Steps 706 through 710 are repeated until heating and heat diffusion have been simulated on each layer of nodes (712). For example, steps 706 through 710 are looped until the last layer is built. The temperature of each node at each time step is recorded in a vector T, which contains the thermal history of the part. Then, a representation of the estimated heat distribution within the AM part is output (714). For example, the computing system can generate a three-dimensional heat map of the estimated heat flux within the AM part, e.g., such as those depicted in FIG. 9. Improved Simulation Procedures using Discrete Green ’s Function
  • Boundary heat loss is directly incorporated into the spectral graph (SG) method.
  • the SG method directly treated only insulated (type 2) boundaries, and boundary heat loss was included by a separate procedure, external to the spectral graph method (step 708), through which the precision of the heat loss was controlled by using multiple small timesteps.
  • the improved SG method embodies a generalized boundary condition that provides for heat loss of any desired level which is equally precise for any time step size. This improves the computational efficiency of the SG method.
  • the SG method 700 is expanded to treat additional ways to add heat to the simulation (step 706).
  • the previous SG method could only simulate heat added at the initial condition.
  • a discrete Green's function GF is defined, which allows for heat addition: from spatial variation of the initial condition (as previously); from time and space variation of volume heating (i.e. heat introduced inside the body as by chemical/nuclear reaction, electric resistance heating, laser absorption, etc.); and, from time and space variation of heating at boundaries of type 1 (specified temperature), type 2 (specified heat flow) or type 3 (specified convective condition).
  • the SG method is improved with weight factors based on the physics of heat flow (step 704d).
  • the previous SG method contained weight factors which were drawn from an image-processing application, consequently calibration of two parameters is required to obtain quantitatively correct results.
  • the new weight factors based on the physics of heat flow between nodes, reduces the number of calibration factors from two to one, thereby simplifying the calibration process, and making the method easier to use.
  • the boundary value problem for temperature is recast into an integral expression containing the GF multiplied by each of the heating effects present in the problem, such as the initial condition, the internal heating function, and any boundary heating functions .
  • the GF method is flexible and powerful because one GF allows for a large family of solutions to be treated in a straight- forward manner.
  • the discrete GF approach is presented by analogy with the continuous GF method. The starting point is the boundary value problem for temperature:
  • n m represents the outward normal vector on the m th portion of the body surface, each of which is characterized by convection coefficient h m [Wm -2 ⁇ -1 ] and local ambient temperature T mm .
  • Material properties are conductivity k and diffusivity This problem will be made dimensionless with the following variables:
  • the temperature is driven by three causative functions: internal heating initial condition T o ; and, boundary heating (B m T mm + q m ). (e.g., improvements to Step 706).
  • the boundary heating is a generalized condition which provides for three types of boundary heating conditions depending on the values of parameters B m and q m .
  • For the nonhomogeneous type 1 boundary divide Eq. (12) by B m and take the limit as B m ⁇ oo, as follows:
  • T in is the temperature contribution from the initial condition, is from internal (volumetric) heating, and T bc is from the nonhomogeneous boundary conditions.
  • Table 1 the expression for each of these three terms is given for the analytical GF solution and by analogy, for the discrete GF solution. The entries in Table 1 will be discussed one at a time.
  • Table 1 Temperature expressions with the continuous GF and with the discrete GF, with contributions to temperature caused by: initial condition internal heating and, boundary conditions
  • the first row of Table 1 shows the contribution to temperature from a nonzero initial condition. To discern the discrete temperature expression from the continuous temperature expression, the spatial integral in the continuous temperature expression is replaced by the discrete GF matrix multiplied by the initial temperature vector ? 0 . This matrix multiplication insures that all nodes with non-zero initial condition have an impact on the resulting temperature. [0082]
  • the second row of Table 1 gives the contribution to the temperature caused by internal heat generation, and as for the first row, the spatial integral from the continuous temperature expression is replaced by the GF matrix multiplied by the causative effect, this time the internal heating vector g to produce the discrete temperature expression. Vector g may vary in space and in time.
  • the third row of Table 1 gives the contribution to temperature for heating at the boundary.
  • the continuous GF method there is a surface integral involving the GF evaluated at the boundary , multiplied by the boundary heat flux This surface integral is developed from a volume integral using Green’s theorem.
  • the discrete GF can only be evaluated at node locations, which may or not be located at the boundary.
  • the surface integral from the continuous temperature expression is replaced by the GF matrix multiplied by an n-vector g m whose elements are non-zero only at boundary nodes where heating takes place.
  • the elements of vector g m are given by: which represent the equivalent volumetric heating caused by heating at boundary nodes. That quantity g m should contain the ratio A m / V m can be demonstrated by equating the energy added to a node by volume generation g to that added by boundary heat flux
  • the use of the GF provides a comprehensive and systematic approach for a family of solutions for a variety of heating conditions, once the GF is known.
  • the discrete GF is developed from the spectral graph method.
  • the Green’s function will be found as the solution to an initial-condition problem with homogeneous boundary conditions.
  • the spectral graph method will be applied to the heat equation, by replacing the Laplacian operator (V 2 ) with a discrete operator called the Laplacian matrix (L).
  • the continuous temperature is replaced by a vector of discrete temperatures at node points in the domain.
  • the discrete form of the heat diffusion equation, with homogeneous boundary conditions and specified initial condition may be written as:
  • the next step is to solve an eigenvalue problem using standard matrix methods.
  • Laplacian matrix L satisfies the following eigenvalue equation: where 0 is the orthogonal eigenvector matrix, A is the diagonal eigenvalue matrix.
  • the eigenvector matrix is orthogonal because L is symmetric and diagonally dominant. Since for an orthogonal matrix the transpose is equal to its inverse, the product of the eigenvector matrix and its transpose is the identity matrix. That is:
  • the spatial behavior is embodied in the eigenfunction matrix cp and the time- evolution behavior is embodied in the eigenvalue matrix A.
  • many eigenvalues within the matrix exponential contribute to the temperature distribution.
  • the temperature depends only on the smallest non-zero eigenvalue (also called the Fiedler eigenvalue), which determines the rate at which the temperature decays toward the final (steady) value. This is analogous to the effect of time on the number of series terms needed from a Fourier-series solution of the continuum heat diffusion equation.
  • the above solution is compared to that from the GF approach.
  • the discrete GF satisfies a (discrete) boundary value problem with homogeneous boundary conditions and impulsive initial condition.
  • One column of the GF matrix contains the temperature response to a unitimpulse initial condition at one node; all the columns together provide the comprehensive response.
  • the required Laplacian matrix is constructed from the node equations for the discrete form of the heat conduction.
  • the node equations are found from an energy balance using the finite-volume theory. A node at the boundary is examined with convection heat loss, and then the result for a non-boundary (interior) node is a straightforward special case.
  • edge weights are non-zero only for near-neighbor nodes, and they depend only on the distance between nodes. More information on edge weights is given later in Section 2.4.
  • the thermal resistance is the sum of two thermal resistors, shown in Fig. 1 : where h L is the heat transfer coefficient is the area for boundary heat transfer [m 2 ], c L [m] is distance from node / to the boundary as shown in Fig. 1, and k is thermal conductivity Heat transfer coefficient hi is an effective value that includes both convective and radiative contributions. Replace the two types of heat flow into Eq. (31) to find the boundary node heat balance: where n is the number of nodes in the body. Although the sum in the above expression is shown over all the nodes in the body, weights w t j are non-zero only for near-neighbor nodes.
  • Ei is a dimensionless conductance for heat loss when node i is located at a type 3 boundary. Note the sign change on this boundary term, which comes from the definition of the normalized temperature. Finally, separate those terms involving neighbor temperatures from the temperature at the /th node:
  • the labels ’diagonal’ and ’off diagonal’ identify the locations of these terms in the /th row of the Laplacian matrix when the /th node is at a type 3 boundary.
  • the Laplacian matrix is assembled from the energy balance relations at each node.
  • the Laplacian matrix is the discrete matrix that replaces the spatial derivatives when the continuous heat equation is replaced by the discrete heat equation.
  • the discrete heat equation is given by Eq. (28):
  • the edge weights developed here are consistent with the finite volume method. For heat transfer in a solid body, the heat flow rate from node j to node i is given by:
  • the heat flow rate depends on the conductivity k [W m 1 K 1 1. the area for heat flow 4 ;/ [m 2 ], and the distance between the nodes dy [m], Then the edge weights are given by
  • the normalized edge weight is constructed using the normalized variables in Eq. (10), where Vt is the (normalized) small volume associated with node i. To fully specify the edge weight, geometric information on the nodal grid is required to determine ratio Ay /Vi. Three grid geometries are discussed below.
  • the node spacing and the heat transfer area is the same for every pair of adjacent nodes.
  • edge weights are identical to the temperature coefficients used in the finite difference method, which means that the one-dimensional SG method and the onedimensional finite difference method have the same spatial behavior. 2.4.2 3D uniform rectangular grid.
  • the edge weight depends upon the details of the geometric relationships among all of the nodes surrounding node i. In the spectral graph method, however, it is important that the edge weights depend primarily on the intemodal distance, rather than on geometric details. In the authors’ previous work, the edge weights for the 3-D random grid had an exponential form, drawn from image processing applications, as follows:
  • Quantity o is the standard deviation of all lengths
  • Quantity f is the gain factor and r n is the neighbor radius, and these two quantities need to be chosen through a calibration process for the method to provide good results. This approach provided reasonable precision with very low computation cost for mesh generation.
  • edge weights for the 3-D random grid were sought that are based on the physics of the problem yet were compatible with the spectral graph method. Edge weights were sought that would: depend on intemodal distance build upon our experience with exponential weights used previously; avoid dependence on local ratio Aij/Vi, and; reduce to 1/d?- in the limit as the random grid moves toward a uniform rectangular grid. [0101] This last requirement suggested that a simple yardstick was needed to determine when a given grid deviates from the uniform rectangular grid. The average distance between adjacent nodes is defined
  • Quantity f may be viewed as the width of a cube containing the average nodal volume; for a uniform rectangular grid f is the exactly the distance between nearest nodes. With quantity f, the following edge weights satisfy the above constraints:
  • edge weights have several important features.
  • the alternate physics-based edge weight performs similarly to (44), and is:
  • the alternate physics-based edge weight (50.1) has similar features to Eq. (50).
  • the edge weight has functional form 1/E 2 which is in agreement with the energy-conserving finite volume formulation.
  • the second and third features are identical to those described above in referenced to Eq. (50).
  • Quantity E) for boundary heat loss must also be determined, which depends upon the nodal surface area for external heat loss A m .
  • a m nodal surface area for external heat loss
  • an average external surface area is used, the same for each boundary node, defined by the overall surface area of the body, divided by the number of surface nodes. This is congruent with the goal of the present work for rough and rapid thermal simulations, as distinct from FE solutions, which require burdensome meshing calculations.
  • This approach is most accurate for bodies with high node counts and generous fillets, and the level of approximation increases as the node count decreases and the fillet radius decreases.
  • the spectral graph method has been extended to provide for internal heating, with heating at the boundary treated as a special case of internal heating.
  • the temperature expression for internal heating contains a time integral, which is discussed here to demonstrate that the spectral graph method is discrete in space and analytic in time.
  • time integral The ease or difficulty in evaluating this time integral depends on the time behavior of internal heating function g.
  • time invariant internal heating then the time integral may be evaluated in closed form. Recall that the eigenvector matrix 0 is a function of space, not of time. Then eigenvector matrix 0 and its transpose may be removed from the time integral, and the time integral may be evaluated as follows:
  • the temperature expression is the sum of a steady part and a complementary transient part. Consequently, the above solution does not apply if the steady solution does not exist, for example, if all the boundaries are insulated (Neumann type). In this circumstance the smallest eigenvalue is zero so that the inverse of the eigenvalue matrix (A-l) does not exist. There are techniques for dealing with this zero-eigenvalue problem which will not be discussed here in the interest of brevity.
  • a closed-form solution may also be found for heating that is piecewise constant in time.
  • This solution may be used as a building block to construct the response to any piecewisein-time heating function, which has application for simulation of a variety of manufacturing processes such as laser welding.
  • T ss is the steady state portion of the solution and T ct is the complementary transient portion of the solution.
  • the steady-state solution is piecewise in space.
  • FIG. 12 shows a schematic of Example 1, the slab with piecewise internal heating, one insulated boundary, and one convection boundary.
  • the errors for the SG method are smaller than those of FD method by about an order of magnitude even at large times after the FD errors have decreased over time.
  • the SMAPE results for several temperature histories are listed in Table 2.
  • the error for the FD method has no clear trend as nx increases. All of the errors in Table 2 are small, less than 0.5 % for the SG method and less than 4% for the FD method.
  • SMAPE Symmetric mean absolute percentage error
  • Example 2 is the temperature in a slab body which is suddenly heated on one boundary and cooled by convection on the other boundary, as follows
  • Figures 8a and 8b show the temperature
  • Figs. 8c and 8d show the relative error
  • the numerical results are very close to the exact values, even though the grid is coarse.
  • the numerical methods have the highest error at early time and at the unheated boundary, and the SG method has smaller errors, by more than an order of magnitude, than the FD method under these conditions.
  • SMAPE Symmetric mean absolute percentage error
  • the method is applied to third simulation example of a parallelepiped with piecewise initial condition for which an exact analytical solution is available for verification. First the exact solution is given, and then the spectral graph method is applied to nodes distributed in a uniform grid and also with a block-random grid.
  • the dimensionless temperature in the parallelepiped is given by:
  • the spectral graph method was carried out for parallelepiped bodies which were initially hot over a small region No calibration is needed for the uniform rectangular grid.
  • the temperature was tracked at three points in each body, identified in FIG. 19 shows locations for observing temperature in the parallelepiped of the third Example simulation case.
  • location (1) is at the center of the initially heated region
  • location (2) is at an interior face of the initially- heated region
  • location (3) is outside the initially heated region.
  • Parameters used for the SG solution were 1536 nodes per unit volume and 80 timesteps.
  • Figure 11 shows that the SG solution agrees very closely with the exact solution.
  • the temperature at locations (1) and (2) fall more rapidly, and further towards zero.
  • the temperature At location (3) the temperature first rises then falls, and for higher Biot number the peak temperature is lower.
  • At large time (not shown) all the temperatures approach zero, and the time it takes to reach zero temperature decreases as the Biot number increases. That is, the cool down is faster at higher heat loss.
  • Table 6 Number of nodes in a parallelepiped body created by n b blocks along each axis and n g nodes within each block. Total number of nodes
  • the calibration procedure for obtaining the gain factor involves a data- fitting procedure between the SG method and the exact solution (refer to Section 2.4.3).
  • the benchmark data for the comparison is the time history of the temperature from the exact solution in the time range (0,0.2) at the center of the initially heated region for a cubic body. Twenty temperature values at uniformly-spaced time points were used.
  • the data fitting procedure was the minimization of the sum-of-square error between the exact solution and the SG model carried out with a Gauss-Newton method. The method converged to four-digit precision in about six iterations and the resulting gam factor was not sensitive to the initial guess.
  • the calibration was earned out for several block-random grids at several Biot numbers. Ten block-random grids were studied for each node density so that averages and variance could be found.
  • the resulting average gain factor is plotted in FIG. 21 versus node density (n/vol) and the variances are shown as error bars.
  • FIG. 21 shows that the gain factor resides in a narrow band of values in the range (0.48 - 0.67), and has no clear trend as node density n/vol varies.
  • FIG. 22 shows temperature versus time for the SG method on a single block-random grid with node density 1536 (nodes per unit volume) compared to the exact solution at three locations shown in FIG. 19 and for three Biot numbers.
  • Parameters used for the SG solution were 1536 nodes on a block-random grid and 80 timesteps.
  • the RMSE errors are included because SMAPE errors skew large at large Biot number, because of division by very small temperature values at later times.
  • the additive manufacturing build was created on an open architecture LPBF system at Edison Welding Institute, Columbus, Ohio.
  • a long wave infrared (LWIR) thermal camera was placed off-axis with an angle about 80° to the horizontal.
  • a representative schematic along with an image of the experimental setup is in shown in FIG. 23.
  • the Micro Epsilon model TIM-640 LWIR thermal camera used in the experiment has a resolution of 640 by 480 pixels. At the cameras height, the spatial resolution of the build plate was approximately 20 pixels per mm 2 .
  • the camera was calibrated according to a black-body technique. This calibration technique enabled the thermal camera to accurately measure top surface temperatures up to 550°C.
  • the parts in the experiment were made from the Inconel 718 powder. Seventeen parts were created from six different geometries, each with a different purpose. For the purposes of this work, data from two inverted half cones were studied with base height 6 mm, base radius 4 mm, and part height 20 mm. These two geometries as well as the completed build plate containing these parts are shown in FIG. 24.
  • FIG. 24 shows the inverted half cone geometry 2400 used to validate this work.
  • two similar geometries were created, one with overhang angle 40 degrees C40 and one with 45 degrees C45.
  • Completed build plate 2402 shows the two inverted half cones (C40, C45) used in this work. Other parts on this build plate were used for a different research topic. These two geometries with overhang were selected as they were expected to experience significant overheating, which can lead to superelevation and build failure in LPBF. For this reason, rapid prediction of the thermal history is of interest.
  • the end-of-cycle surface temperature was extracted in the following manner.
  • the raw temperature has three prominent features, demarcated (A), (B), and (C), which correspond to specific process events (illustrated in FIG. 25E).
  • the thermal camera acquires data only when the laser is active through a triggering mechanism.
  • the first large spike marked (A) is when the laser is striking the sampled 4 mm 2 pixel region.
  • the temperature recorded at (B) is momentarily interrupted at the time the laser and camera are both switched off.
  • the epoch marked (C) and beyond is for the next layer processed by the laser.
  • the recoater fetches powder, and a fresh powder layer is deposited.
  • the time for recoating is measured to be 11 seconds, and remains fixed irrespective of the process conditions or number of parts on the build plate.
  • the temperature in the instant just before the laser strikes the sampled area again, before the melting of a new layer, is termed as the end-of-cycle surface temperature.
  • Plotted in FIG. 25D is the end-of-cycle surface temperature for the 9 pixel by 9 pixel area (4 mm 2 ) of the cone-shaped part C45 sampled in FIG. 25A.
  • the end-of-cycle temperature from one cycle is the initial condition for the body in the subsequent heating cycle, except for the newly -fused surface layer which increases the size of the computational domain.
  • the temperature problem must be linear, so that the material properties must not be functions of the temperature. In practice this means that the properties are evaluated at an appropriate average value during each heating cycle. To maintain symmetry of the Laplacian matrix the edge weights must be symmetric consequently spatial variation in the material properties are ruled out.
  • the newly-fused layer is added at the metal fusion temperature; this temperature and the size of the new layer determines the amount of thermal energy added in each heating cycle.
  • Every numerical method contains parameters that must be chosen so that the results are physically meaningful.
  • calibration is needed to find the gain factor using the method described in Section 2.4.3.
  • the analytical solution described in Section 3.3 was used to compute a single heating/cooling cycle in an Inconel 718 parallelepiped depicted in Figure 9 with each side of length 25 mm.
  • the spectral graph method was applied to this geometry, and the gain factor was chosen to provide the best fit between the analytical solution and the spectral graph model.
  • Other parameters used in the calibration are given in Table 8.
  • Table 8 Summary of simulation of parameters used for the gain factor calibration and inverted half cone simulations.
  • Another calibration step is to determine the correct level of heat loss for the simulation.
  • the SG model was applied to inverted-cone geometry C40, with the larger overhang, and end-of-cycle temperatures were compared to data from the experimental build.
  • the model was run with 50 superlayers, that is, 50 heating cycles in succession to represent the actual 500 layers in the build.
  • the build plate on which the part is built, the primary heat sink in the problem was modeled as a large convection coefficient (also called a contact conductance).
  • the value of h b 300 W/(m 2 K) was chosen for the build plate; this value is sufficiently large to represent a fixed-temperature boundary condition, because larger hb values give the same result.
  • the effect of heat loss to the surrounding powder was modeled as another convection coefficient, to describe the heat loss to the low-conductivity metal powder on the sides of the fused part, and heat loss to the gas at the exposed upper surface of the fused part.
  • SMAPE error between the SG model for the C40 part and the end-of-cycle temperature obtained from the LWIR thermal camera was minimized.
  • h w 0. 1 [W/m -2 K ⁇ r ] was chosen; other parameters for the 50-layer simulation are provided in Table 8.
  • the SG model for the calibration case is shown in FIG. 26 graph 2600.
  • the dashed red lines are the ensemble average SG model values and the shaded region show the variance from ten block-random grids.
  • the solid black lines are the experimentally measured values for each geometry.
  • the dashed line shows the ensemble average of 10 computer runs using 10 different block-random grids and the shaded region shows the variance of the 10 different grids.
  • the average error is less than 3% SMAPE and 29 degrees C RMSE.
  • This mismatch is primarily a function of the overall energy budget in each heating cycle, that is, heat in minus heat out.
  • areas for future work include improving heat addition in the simulation to more closely agree with the physics of laser absorption, and, improving heat loss to the surroundings, for example by including radiation heat loss.
  • the SG method has been extended to directly incorporate heat loss with a generalized boundary condition, which is a distinct improvement over the ad hoc heat loss method used previously. Improved edge weights in the Laplacian matrix are now based on the physics of the problem, which reduces the number of calibration parameters and consequently simplifies the calibration process.
  • the improved SG method has been used to develop a discrete Green’s function for comprehensive treatment of several heating effects: space-varying initial conditions; time-and-space varying boundary heating; and, time-and-space varying internal heating.
  • the precision of the method was determined from one-dimensional and three dimensional benchmark examples for which exact solutions were available. The one- dimensional examples were also compared with finite difference solutions.
  • the method was also applied to a 50-layer simulation of an additive manufacturing process for two bodies with severe overhang, and the simulation results were validated by comparison with experimental temperature measurements.
  • the model agrees with the experiments within 5% SMAPE with model computation time of less than one minute on a desktop computer.
  • the rapid computation time of this improved thermal model provides an opportunity for application to flaw detection using real-time thermal sensor data; this is an area of ongoing research.
  • the computational advantage of the SG method applied to additive manufacturing comes from a combination of low-cost grid generation, only one eigenvalue solution to find the GF, and semi-analytic behavior on time which allows for time steps of any size.
  • the finite element (FE) method has high-cost mesh generation and numerical time- integration with many small time steps required to control precision.
  • the computational advantage of the SG method is strong for the rough and rapid thermal simulations that are presently sufficient to advance the field of thermal modeling of 3D printing.
  • the computational cost of the eigenvalue problem in the SG method scales as O(n 3 ), compared to the computational cost of the FE method which scales as O(m n 2 ) where n is the number of nodes and m is the number of timesteps.
  • O(n 3 ) the computational cost of the FE method which scales as O(m n 2 ) where n is the number of nodes and m is the number of timesteps.
  • FIG. 28 depicts an example computing system, according to implementations of the present disclosure.
  • the system 2800 may be used for any of the operations described with respect to the various implementations discussed herein.
  • the system 2800 may include one or more processors 2810, a memory 2820, one or more storage devices 2830, and one or more input/output (I/O) devices 2850 controllable via one or more I/O interfaces 2840.
  • the various components 2810, 2820, 2830, 2840, or 2850 may be interconnected via at least one system bus 2860, which may enable the transfer of data between the various modules and components of the system 2800.
  • the processor(s) 2810 may be configured to process instructions for execution within the system 2800.
  • the processor(s) 2810 may include single-threaded processor(s), multi -threaded processor(s), or both.
  • the processor(s) 2810 may be configured to process instructions stored in the memory 2820 or on the storage device(s) 2830. For example, the processor(s) 2810 may execute instructions for the various software module(s) described herein.
  • the processor(s) 2810 may include hardwarebased processor(s) each including one or more cores.
  • the processor(s) 2810 may include general purpose processor(s), special purpose processor(s), or both.
  • the memory 2820 may store information within the system 2800.
  • the memory 2820 includes one or more computer-readable media.
  • the memory 2820 may include any number of volatile memory units, any number of nonvolatile memory units, or both volatile and non-volatile memory units.
  • the memory 2820 may include read-only memory, random access memory, or both.
  • the memory 2820 may be employed as active or physical memory by one or more executing software modules.
  • the storage device(s) 2830 may be configured to provide (e.g., persistent) mass storage for the system 2800.
  • the storage device(s) 2830 may include one or more computer-readable media.
  • the storage device(s) 2830 may include a floppy disk device, a hard disk device, an optical disk device, or a tape device.
  • the storage device(s) 2830 may include read-only memory, random access memory, or both.
  • the storage device(s) 2830 may include one or more of an internal hard drive, an external hard drive, or a removable drive.
  • One or both of the memory 2820 or the storage device(s) 2830 may include one or more computer-readable storage media (CRSM).
  • the CRSM may include one or more of an electronic storage medium, a magnetic storage medium, an optical storage medium, a magneto-optical storage medium, a quantum storage medium, a mechanical computer storage medium, and so forth.
  • the CRSM may provide storage of computer- readable instructions describing data structures, processes, applications, programs, other modules, or other data for the operation of the system 2800.
  • the CRSM may include a data store that provides storage of computer-readable instructions or other information in a non-transitory format.
  • the CRSM may be incorporated into the system 2800 or may be external with respect to the system 2800.
  • the CRSM may include read-only memory, random access memory, or both.
  • One or more CRSM suitable for tangibly embodying computer program instructions and data may include any type of non-volatile memory, including but not limited to: semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks.
  • the processor(s) 2810 and the memory 2820 may be supplemented by, or incorporated into, one or more applicationspecific integrated circuits (ASICs).
  • ASICs applicationspecific integrated circuits
  • the system 2800 may include one or more I/O devices 2850.
  • the I/O device(s) 2850 may include one or more input devices such as a keyboard, a mouse, a pen, a game controller, a touch input device, an audio input device (e.g., a microphone), a gestural input device, a haptic input device, an image or video capture device (e.g., a camera), or other devices.
  • the I/O device(s) 2850 may also include one or more output devices such as a display, LED(s), an audio output device (e.g., a speaker), a printer, a haptic output device, and so forth.
  • the I/O device(s) 2850 may be physically incorporated in one or more computing devices of the system 2800, or may be external with respect to one or more computing devices of the system 2800.
  • the system 2800 may include one or more I/O interfaces 2840 to enable components or modules of the system 2800 to control, interface with, or otherwise communicate with the I/O device(s) 2850.
  • the I/O interface(s) 2840 may enable information to be transferred in or out of the system 2800, or between components of the system 2800, through serial communication, parallel communication, or other types of communication.
  • the I/O interface(s) 2840 may comply with a version of the RS-232 standard for serial ports, or with a version of the IEEE 1284 standard for parallel ports.
  • the I/O interface(s) 2840 may be configured to provide a connection over Universal Serial Bus (USB) or Ethernet.
  • USB Universal Serial Bus
  • the I/O interface(s) 2840 may be configured to provide a serial connection that is compliant with a version of the IEEE 1394 standard.
  • the I/O interface(s) 2840 may also include one or more network interfaces that enable communications between computing devices in the system 2800, or between the system 2800 and other network-connected computing systems.
  • the network interface(s) may include one or more network interface controllers (NICs) or other types of transceiver devices configured to send and receive communications over one or more communication networks using any network protocol.
  • NICs network interface controllers
  • Computing devices of the system 2800 may communicate with one another, or with other computing devices, using one or more communication networks.
  • Such communication networks may include public networks such as the internet, private networks such as an institutional or personal intranet, or any combination of private and public networks.
  • the communication networks may include any type of wired or wireless network, including but not limited to local area networks (LANs), wide area networks (WANs), wireless WANs (WWANs), wireless LANs (WLANs), mobile communications networks (e.g., 3G, 4G, Edge, etc.), and so forth.
  • the communications between computing devices may be encrypted or otherwise secured.
  • communications may employ one or more public or private cryptographic keys, ciphers, digital certificates, or other credentials supported by a security protocol, such as any version of the Secure Sockets Layer (SSL) or the Transport Layer Security (TLS) protocol.
  • SSL Secure Sockets Layer
  • TLS Transport Layer Security
  • the system 2800 may include any number of computing devices of any type.
  • the computing device(s) may include, but are not limited to: a personal computer, a smartphone, a tablet computer, a wearable computer, an implanted computer, a mobile gaming device, an electronic book reader, an automotive computer, a desktop computer, a laptop computer, a notebook computer, a game console, a home entertainment device, a network computer, a server computer, a mainframe computer, a distributed computing device (e.g., a cloud computing device), a microcomputer, a system on a chip (SoC), a system in a package (SiP), and so forth.
  • SoC system on a chip
  • SiP system in a package
  • a computing device may include one or more of a virtual computing environment, a hypervisor, an emulation, or a virtual machine executing on one or more physical computing devices.
  • two or more computing devices may include a cluster, cloud, farm, or other grouping of multiple devices that coordinate operations to provide load balancing, failover support, parallel processing capabilities, shared storage resources, shared networking capabilities, or other aspects.
  • Implementations and all of the functional operations described in this specification may be realized in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations may be realized as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus.
  • the computer readable medium may be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more of them.
  • the term “computing system” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers.
  • the apparatus may include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.
  • a propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal that is generated to encode information for transmission to a suitable receiver apparatus.
  • a computer program (also known as a program, software, software application, script, or code) may be written in any appropriate form of programming language, including compiled or interpreted languages, and it may be deployed in any appropriate form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment.
  • a computer program does not necessarily correspond to a file in a file system.
  • a program may be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code).
  • a computer program may be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.
  • the processes and logic flows described in this specification may be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output.
  • the processes and logic flows may also be performed by, and apparatus may also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).
  • FPGA field programmable gate array
  • ASIC application specific integrated circuit
  • processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any appropriate kind of digital computer.
  • a processor may receive instructions and data from a read only memory or a random access memory or both.
  • Elements of a computer can include a processor for performing instructions and one or more memory devices for storing instructions and data.
  • a computer may also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks.
  • mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks.
  • a computer need not have such devices.
  • a computer may be embedded in another device, e.g., a mobile telephone, a personal digital assistant (PDA), a mobile audio player, a Global Positioning System (GPS) receiver, to name just a few.
  • Computer readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks.
  • the processor and the memory may be supplemented by, or incorporated in, special purpose logic circuitry.
  • implementations may be realized on a computer having a display device, e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor, for displaying information to the user and a keyboard and a pointing device, e.g., a mouse or a trackball, by which the user may provide input to the computer.
  • a display device e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor
  • keyboard and a pointing device e.g., a mouse or a trackball
  • Other kinds of devices may be used to provide for interaction with a user as well; for example, feedback provided to the user may be any appropriate form of sensory feedback, e.g., visual feedback, auditory feedback, or tactile feedback; and input from the user may be received in any appropriate form, including acoustic, speech, or tactile input.
  • Implementations may be realized in a computing system that includes a back end component, e.g., as a data server, or that includes a middleware component, e.g., an application server, or that includes a front end component, e.g., a client computer having a graphical user interface or a web browser through which a user may interact with an implementation, or any appropriate combination of one or more such back end, middleware, or front end components.
  • the components of the system may be interconnected by any appropriate form or medium of digital data communication, e.g., a communication network. Examples of communication networks include a local area network (“LAN”) and a wide area network (“WAN”), e.g., the Internet.
  • LAN local area network
  • WAN wide area network
  • the computing system may include clients and servers.
  • a client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other.
  • This specification contains many specifics, these should not be construed as limitations on the scope of the disclosure or of what may be claimed, but rather as descriptions of features specific to particular implementations. Certain features that are described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation may also be implemented in multiple implementations separately or in any suitable sub-combination.

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Abstract

L'invention concerne des procédés, des systèmes et un appareil, y compris des programmes informatiques codés sur un support de stockage informatique, pour un procédé de simulation de transfert thermique de fabrication d'additif. Le procédé comprend la conversion d'un modèle d'un objet en une représentation de nœud de l'objet et la génération d'un graphe de réseau de l'objet sur la base de la représentation de nœud. Pour chaque bloc de nœuds dans la représentation de nœud, le procédé comprend : l'application d'une chaleur simulée au bloc de nœuds par de multiples fonctions de causalité, la réalisation d'un bilan énergétique du flux de chaleur entrant dans le nœud et sortant du nœud pour déterminer l'énergie stockée dans le nœud, et l'estimation d'une diffusion de chaleur vers d'autres nœuds à l'aide de poids de bord basés sur la physique entre des nœuds dans le graphique de réseau. Le procédé comprend la génération d'une représentation d'une distribution de chaleur estimée à l'intérieur de l'objet.
PCT/US2022/048296 2021-10-29 2022-10-28 Modélisation thermique de fabrication d'additif Ceased WO2023076636A1 (fr)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117910099A (zh) * 2023-12-29 2024-04-19 同济大学 一种适用于3d打印墙体的各向异性热传递分析方法及系统

Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20160246908A1 (en) * 2014-09-19 2016-08-25 Siemens Product Lifecycle Management Software Inc. Computer-aided simulation of additive manufacturing processes
US20170337307A1 (en) * 2016-05-20 2017-11-23 Dassault Systemes Simulia Corp. Scalable finite element simulation of additive manufacturing
WO2020056388A1 (fr) * 2018-09-13 2020-03-19 Board Of Regents Of The University Of Nebraska Simulation d'un flux de chaleur dans une fabrication additive
WO2020131053A1 (fr) * 2018-12-19 2020-06-25 Hewlett-Packard Development Company, L.P. Détermination d'une empreinte thermique d'une pièce imprimée en trois dimensions
US20200327267A1 (en) * 2019-04-12 2020-10-15 Iowa State University Research Foundation, Inc. Interstitial control during additive manufacturing
WO2021097248A1 (fr) * 2019-11-14 2021-05-20 University Of Washington Rétroaction à boucle fermée pour simulation de fabrication additive

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20160246908A1 (en) * 2014-09-19 2016-08-25 Siemens Product Lifecycle Management Software Inc. Computer-aided simulation of additive manufacturing processes
US20170337307A1 (en) * 2016-05-20 2017-11-23 Dassault Systemes Simulia Corp. Scalable finite element simulation of additive manufacturing
WO2020056388A1 (fr) * 2018-09-13 2020-03-19 Board Of Regents Of The University Of Nebraska Simulation d'un flux de chaleur dans une fabrication additive
WO2020131053A1 (fr) * 2018-12-19 2020-06-25 Hewlett-Packard Development Company, L.P. Détermination d'une empreinte thermique d'une pièce imprimée en trois dimensions
US20200327267A1 (en) * 2019-04-12 2020-10-15 Iowa State University Research Foundation, Inc. Interstitial control during additive manufacturing
WO2021097248A1 (fr) * 2019-11-14 2021-05-20 University Of Washington Rétroaction à boucle fermée pour simulation de fabrication additive

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
FOTEINOPOULOS PANAGIS; PAPACHARALAMPOPOULOS ALEXIOS; STAVROPOULOS PANAGIOTIS: "On thermal modeling of Additive Manufacturing processes", CIRP JOURNAL OF MANUFACTURING SCIENCE AND TECHNOLOGY, vol. 20, 1 January 1900 (1900-01-01), AMSTERDAM, NL , pages 66 - 83, XP085316450, ISSN: 1755-5817, DOI: 10.1016/j.cirpj.2017.09.007 *

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117910099A (zh) * 2023-12-29 2024-04-19 同济大学 一种适用于3d打印墙体的各向异性热传递分析方法及系统

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