A Generalized Feed Forward Dynamic Adaptive Mesh ...

Nachiket Patil 3DSIM, LLC, 201 E. Jefferson Street, Louisville, KY 40202 e-mail: [email protected]

Deepankar Pal1 Assistant Professor Department of Mechanical Engineering, University of Louisville, Louisville, KY 40292; J.B. Speed School of Engineering, University of Louisville, Louisville, KY 40292 e-mail: [email protected]

H. Khalid Rafi Department of Industrial Engineering, University of Louisville, Louisville, KY 40292; J.B. Speed School of Engineering, University of Louisville, Louisville, KY 40292 e-mail: [email protected]

Kai Zeng Department of Industrial Engineering, University of Louisville, Louisville, KY 40292; J.B. Speed School of Engineering, University of Louisville, Louisville, KY 40292 e-mail: [email protected]

Alleyce Moreland Mound Laser and Photonics Center, 2941 College Drive, Kettering, OH 45420

Adam Hicks Mound Laser and Photonics Center, 2941 College Drive, Kettering, OH 45420

David Beeler

A Generalized Feed Forward Dynamic Adaptive Mesh Refinement and Derefinement Finite Element Framework for Metal Laser Sintering—Part I: Formulation and Algorithm Development A novel multiscale thermal analysis framework has been formulated to extract the physical interactions involved in localized spatiotemporal additive manufacturing processes such as the metal laser sintering. The method can be extrapolated to any other physical phenomenon involving localized spatiotemporal boundary conditions. The formulated framework, named feed forward dynamic adaptive mesh refinement and derefinement (FFD-AMRD), reduces the computational burden and temporal complexity needed to solve the many classes of problems. The current study is based on application of this framework to metals with temperature independent thermal properties processed using a moving laser heat source. The melt pool diameters computed in the present study were compared with melt pool dimensions measured using optical micrographs. The strategy developed in this study provides motivation for the extension of this simulation framework for future work on simulations of metals with temperature-dependent material properties during metal laser sintering. [DOI: 10.1115/1.4030059]

Mound Laser and Photonics Center, 2941 College Drive, Kettering, OH 45420

Brent Stucker Department of Industrial Engineering, University of Louisville, Louisville, KY 40292; J.B. Speed School of Engineering, University of Louisville, Louisville, KY 40292 e-mail: [email protected]

Introduction 1

Corresponding author. Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received January 29, 2014; final manuscript received March 11, 2015; published online July 8, 2015. Assoc. Editor: Jack Zhou.

The finite element method (FEM) is a widely used numerical method for solving systems of partial differential equations (PDEs). A typical FEM algorithm involves discretizing a domain into a finite number of elements and formulating a weak form of the PDE. A weak form of differential equations can be

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decomposed into a system of linear equations with the help of weighted residuals or the principle of minimum energy. The computational cost when solving these systems of equations and the accuracy of the solution depends on the extent of refinement. A strategy where mesh refinement is decided based on variations across a domain is called adaptive mesh refinement (AMR) [1]. AMR is useful for reducing the computational costs involved in problems with significant spatial variations or localized behavior. Adaptive Meshing. AMR was proposed by Berger and Oliger to solve wave equations with localized steep gradients [1,2]. AMR provides significant advantages in terms of reliability and computational costs for simulating systems with different characteristic spatial lengths. For example, wave propagation leads to local deformation, which requires refinement in that region to capture the localized phenomena. The multiscale nature of AMR enables one to more efficiently solve problems where computational cost and time would be very high if the problem were to be solved with a high resolution everywhere [2]. To apply AMR effectively, acceptable values of error bounds and estimated error are a consideration. The Error Estimates in FEM section explains different error estimation techniques and their formulation in depth. Error Estimates in FEM. Traditionally, convergence of a finite element model is checked using mesh convergence. This method is computationally expensive for time dependent dynamic problems where adaptive meshing is required. General adaptive meshing software tools [3] employ error estimators to estimate accuracy of the solution obtained within a particular level of refinement. There are two types of error estimators available [4], namely, a priori and posteriori estimators. A priori error estimators are used to understand the asymptotic behavior of the errors but are not used to determine an actual error estimate. Posteiori error estimators work on the solution itself to obtain estimates of the actual solution errors. In the following discussion, both of these estimators are elaborated in greater detail. A Priori Error Estimates. A priori error estimates provide useful information on the asymptotic behavior of the approximation. The Lax theorem [5] is the fundamental theorem for a priori error estimation in the numerical solution of PDEs. It starts with a generalized representation of a PDE in the following form, where L is a differential operator and  is an actual solution of the differential equation L ¼ f (1) to understand the Lax theorem assume a system where L ¼

@2 ¼f @x2

(2)

Taylor’s expansion of the above equation using central difference approximation gives @ 2  ði þ 1Þh  2ðihÞ þ ði  1Þh ¼ þ Oðh2 Þ @x2 h2

(3)

Assuming a discrete solution ui ¼ ðihÞ

(4)

where fi ¼ f ðu ¼ ihÞ Consistency equations can be further defined as i ¼ ðihÞ  h  L u i fi ¼ si ¼ Oðh2 Þ

(7)

and the stability equation  h   L 1 

(8)

is bounded and independent of h Thus, error is defined as e¼u

(9)

Lh e ¼ s Convergence is then derived from stability and consistency    kek   Lh 1 s  Cksk

(10)

This theorem proves that the convergence is proportional to h2 for the elliptic differential equation considered here. This is true with respect to linear FEM formulations with bilinear shape functions. Posteriori Error Estimates. In FEM, a domain is decomposed into small elements and a solution of the differential equation in each element is assumed to have the form b  ¼ Ni ðxÞqi

(11)

where Ni are the shape functions and qi is an approximate solution of the differential equation at the ith node of the element. A posteriori error [6] in the approximate solution b  is defined as ðxÞ e ð xÞ ¼ ðxÞ  b

(12)

Similarly, errors in integration point variables, namely, stress ðrÞ and strain ðeÞ are defined as er ð xÞ ¼ rðxÞ  r^ðxÞ ee ð xÞ ¼ eðxÞ  ^eðxÞ

(13)

These error measures are a function of position vector x and can be positive or negative depending on the location of x. A better error indicator should be a scalar so that error criteria can be checked and decisions can be made accordingly. For this purpose, different error norms are used. For example, an L2 norm is defined as ð (14) kk2 ¼ ð xÞð xÞdX X

The energy norm L2 norm of the conjugate stress and strain spaces is one of the most widely applied norms in FEM error analysis of linear elastic structural problems. It is defined as ð ð kk2 ¼ ðe  e ÞEðe  e ÞdX ¼ eTe er E dX (15) z}|{

X

z}|{

X

A discrete operator Lh is defined as Lh u ¼

uiþ1  2ui þ ui1 h2

A relative percentage error norm is defined as (5) n ¼ 100 

The final discretized system of equations will be 

 Lh u i ¼ fi

041001-2 / Vol. 137, AUGUST 2015

(6)

norm of the error ðeÞ norm of the corresponding variable 

(16)

The main challenge in the posteriori error estimate mentioned so far is to derive an exact solution ðxÞ for the differential equation. Transactions of the ASME

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It is not possible in most practical problems involving complex geometries to find an exact analytical solution. It has been shown that in continuous smoothened solutions for stress and strain it is more accurate compared to piecewise approximate solutions determined using FEM. Among various methods of smoothening FEM solutions, super convergent patch recovery [7] of the elements is one of the most accurate methods. In order to reduce the error peaks due to localized variations and steep gradients in material response, various localized refinement methods have been proposed. The refinements are adaptive in nature where the refinement sizes are corrected inversely with error magnitudes. The process of correction of refinement sizes is iterative in nature where the refinements are corrected until the local error magnitudes are below a user-specified error-tolerance. Generally, the error-tolerances are kept as a very small fraction of the initial maximum response magnitude or initial maximum magnitude of the external stimulus or combinations of both depending on the definition of error calculation. When the error is out of the bound, the mesh is refined adaptively to bring it within the tolerance level. Various structured AMR strategies are discussed in Structured Adaptive Meshes. Unstructured AMR strategies have not been discussed here since they are not within the scope of the application. Structured Adaptive Meshes. In structured adaptive meshes, the mapped orthonormality of the grid is strictly maintained or a particular motif is maintained. The basis of the mesh can be regular Cartesian or curvilinear in nature. Block structured mesh refinement [1,2] involves a mesh refinement of multiple sub domains such that two domains can share a boundary but no area can be shared. Each domain can be meshed independently and can be further refined to reach higher amounts of resolution. This strategy can be explained through a tree diagram as shown in Fig. 1. The refinement inside each subdomain creates a mismatch of mesh density at the interface of the coarse grid and refined subdomain grid (Fig. 2) and can lead to erroneous results. This can be avoided with additional constraints at interfaces. The disadvantage of having additional constraints is that the solution may not reach its equilibrium distribution. The irregular nodes in the block structured adaptive mesh can be avoided with an adaptive mesh generated with mesh distortion as shown in Fig. 3. This method has been used along with multizone adaptive grid generation [8] or moving meshes [9]. These methods require additional computations to compute the characteristics of a curvilinear mesh grid, as shown in Fig. 3. Adaptive Meshing in Dynamic Problems and Data Transfer Between Meshes. Certain classes of problems such as large deformation or time dependent dynamic problems require adaptive dynamic mesh refinement where the mesh is dynamically adapted to new boundary conditions, evolving constitutive laws or geometry. This dynamic adaptivity of the mesh calls for data transfer from the parent mesh to the newly adapted mesh [10,11].

Fig. 1 (a) Geometry and (b) data structure of a three-level mesh tree [8]

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Fig. 2 An example of a three-level, block-structured AMR hierarchy [9]

Various interpolation techniques are used for this purpose. Two widely used interpolation techniques [10–16] are, namely, the radial basis function method and the inverse isoparametric method. In the present work, the isoparametric interpolation method [16] is used considering its computational efficiency and compatibility with FEM discretization assumptions. The following discussion, describes a special class of dynamic problems where refinement, derefinement, and data transfer is required frequently. Literature on Dynamic Adaptive Mesh Generation. “Dynamic adaptive meshing” term has been used for various types of adaptive meshing strategies which involve mesh adaptively for dynamic boundary conditions or response evolution. A dynamically adaptive mesh [17] has been prepared with the help of moving mesh strategy involving solution of variational equations derived for moving mesh generation. Similar moving FEM algorithms with various error criterions have been tried in the literature. An error based redistribution [18] and adaptivity in moving mesh has been tried and implemented to solve localized dynamic problems. A level set method [19] is also been used for creating a dynamic moving mesh. In case of spatiotemporal problems, moving mesh requires large number of adaptations, which entail solving the moving mesh equations multiple times. The block structured refinement can provide same computational reduction because it does not require solving moving mesh equations to generate mesh. The disadvantage associated with this methodology is that large number local adaptations are required to follow the moving heat source. The error estimate calculations and graph generation can be avoided in with intelligent schemes where patterns learned [20] from similar problems are used to develop intelligent schemes. Combination the block structured refinement and intelligent adaptive meshing strategy can help in reduction of time required to perform error estimates and matrix assembly up to certain extent but at every iteration stiffness matrices are required to be assembled again. The easier methodology is to solve these problems can be having a refined moving region with upper bounded on refinement. This will avoid repeated error estimate calculations, mesh graph generation, and possibility of intelligent matrix reassembly. A similar moving refinement of the localized regions near a machine tool has been attempted with coded box cell [21] excluding a

Fig. 3

Adaptive mesh with mesh distortion

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derefinement of the refined localized region. In the present work, a new strategy is attempted to generate a moving finite element refinement zone with cut–paste–solve–restore algorithm. The mesh graph generation (excludes stiffness matrix assembly) by cut paste strategy [22] has been attempted for triangulated meshes. The computational time required to generate stiffness matrices for newly adapted mesh graph is significantly higher compared to the new mesh graph generation or modification. This work will also focus on efficient regeneration of stiffness matrix for the new location of the moving refinement box. Literature on FEM Simulations in Metal Laser Sintering. A two-dimensional plain stress model [23] has been attempted to calculate thermal distribution and residual stresses using FEM analysis. Axisymmetric model [24] is attempted with the assumption that the powders transforms into a sphere due to the surface tension. A symmetric boundary condition model [25] with respect to a plane in the vertical direction of scan path has been attempted with consideration of laser light penetration and evaporation. Simulations with a global scale model [26] which averages out the scan pattern into area heat source has been considered to reduce computational complexity. Such a model will calculate thermal evolution and stresses away from the scan pattern more accurately compared to evolution just near the laser spot. A single scale 3D finite element model [27] has been attempted to simulate thermal distribution and residual stresses along with temperature and porosity dependent thermal properties. In the simulation of metal laser sintering or spatiotemporal problems, multiscale simulation is necessary to account for multiple macroscopic scales involved in the analysis. A two scale model for subdomain refinement [28] has been attempted with temperature dependent thermal properties and phase transformation has been attempted. An element birth and death process [29] has been implemented to simulate metal laser sintering process along with material nonlinearities in 3D transient thermal finite element model. The simulation uses a two scale resolution model with tapering in refinement transition zone. A similar mesh simulation [30] has been attempted for overhanging part build in metal laser sintering with the help of ANSYS. Childs et al. [31] investigated the influence of process parameters on the mass of melted single layers in metal laser sintering and found that melted mass increased with increasing scanning speed. A static two scale model [28,32] is prepared for predicting melt pool dimensions for a geometry of 6  9 mm area. Spatiotemporal Periodic Problems and Metal Laser Sintering. A spatiotemporal periodic function is defined as a function which repeats itself over time with rigid translation in space. It can be expressed in the following form:

[33]. The basic PDE and FEM formulation required to solve thermal behavior during metal laser sintering is given in Governing Equations and Finite Element Formulation in FFD-AMRD. Recent Advances in Advanced and Traditional Manufacturing. Lasers have been used for material processing for the purpose of subtractive as well as additive processing. Some of the recent work in laser based material processing or manufacturing metallic as well as nonmetallic materials can be found in literature [34–38]. Relationship between material properties, microstructure and processing has been an active area of research at present [39–44]. Simulation of various processes to compute these nonlinear relations has been also a topic of interest due to its application in optimizing processes for better material properties and improved process control [45–51]. Governing Equations and Finite Element Formulation in FFD-AMRD. The governing heat transfer equation can be written as   @qx @qy @qz @T (19) þ þ þ Q ¼ qc  @t @x @y @z where qx ; qy ; qz are components of heat flow through unit area. According to Fourier’s law @T @x @T qy ¼ k @y @T qz ¼ k @z

qx ¼ k

Element Formulations of 3D Thermal FEM. Equation (20) has been decomposed into a weak form [52] as the following equation:     ½Cthermal  T_ þ ½KThermal fT g ¼ RQ (21) A description of the three matrices in Eq. (21) is given below. (i)

Thermal stiffness formulation The thermal stiffness matrix KThermal is expressed as follows: ð ð ! KThermal ¼ BTThermal k BThermal dV þ hN T NdS (22) V

f ðx; tÞ ¼ f ðx þ vT; t þ T Þ

(17)

Where the v represents the speed and T represents the time-period of the problem. An assumption for continuously spatiotemporal periodic problems in time is given in the following equation: T!0

(18)

Spatiotemporal periodicity can be observed in various problems including metal laser sintering, wave propagations, thermodynamics of welding, various manufacturing processes such as sheet metal rolling, and dynamic tire road surface contact. The present work is focused on simulation of metal laser sintering with using a novel FFD-AMRD finite element algorithm. This algorithm involves a moving fine-mesh box, which can capture the spatiotemporal phenomena shown in Fig. 4. Metal laser sintering is an additive manufacturing process in which the surface of a powder bed is melted layer by layer to create a 3D part with complex geometry. Further detailed descriptions and literature related to our metal laser sintering thermomechanical formulation can be found in a related article 041001-4 / Vol. 137, AUGUST 2015

(20)

S

where BThermal is known as the flux–temperature matrix. The size of this matrix is 3  8. For a brick element comprised of eight integration points as shown in Fig. 5, Eq. (22) is cumulatively repeated for all the integration points leading to KThermal of size 8  8. dV denotes the volume of the element. The surface integral is valid when the bulk is exposed to a convection boundary condition. The surface integral is employed at the exposed surface with twodimensional shape functions [52] and then transformed to three-dimensional space. h in the above mentioned case is the convective heat transfer coefficient which has been assumed to be 12.5 W/(m2K) for Argon [54]. Argon fills the chamber atmosphere for reactive materials processed using metal laser sintering. (ii) Thermal specific heat matrix formulation The thermal specific heat matrix Cthermal is expressed as follows: Cthermal ¼

ð

qN T cNdV

(23)

V

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Fig. 4

Schematic diagram showing FFD-AMRD dynamic mesh

where ½ N  ¼ ½N1 N2 N3 …:: N8  are the three-dimensional nodal shape functions of size 1  8. Cthermal is computed once and it has a size of 8  8. (iii) Thermal flux vector formulation The thermal flux vector RQ of size 8  1 is evaluated as follows: fRQ g ¼

ð S

ðqðr; tÞ  n^ÞN T dS þ

ð

hTe N T dS

(24)

S

where q is the input heat flux. dS denotes the surface area of the element. The second surface integral in Eq. (24) is valid only when the convection boundary conditions operate and hence should be employed on those boundaries only. In this scenario, Te denotes the temperature of the ambient environment.

 Fig. 5 A brick element with eight integration points. The B integration scheme has been incorporated [53].

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Calculation of these matrices and solving Eq. (21) requires significant computational effort if uniform mesh density is employed. In FFD-AMRD Framework for Applications in Spatiotemporally Periodic Localized Boundary Conditions, a proposed efficient numerical algorithm is presented with mathematical error estimation rationality for its development. AUGUST 2015, Vol. 137 / 041001-5

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FFD-AMRD Framework for Applications in Spatiotemporally Periodic Localized Boundary Conditions

element number and eight node numbers in a strictly positive triple product sequence

In this section, the FFD-AMRD algorithm is described and applied to metal laser sintering. First, a mathematical rationale for using refined moving fine mesh is established. The thermal behavior in the case of metal laser sintering can be decomposed into local and global problems in terms of thermal curvature. It can be approximately demonstrated using a cylindrical coordinate system in 2D. Here, thermal flow is assumed to be two-dimesional and the Gaussian heat flux from the laser is assumed to be an external heat input as hðrÞ.

cordð N Þ ¼ ½XN ; YN ; ZN 

k

d2 T dT 1 dT k  hðr Þ ¼ dr 2 dt r dr

(25)

The laser is centered at r ¼ 0 with Gaussian flux. The dT/dr term in the above equation has a magnitude on the order of (1/r2) provided the integral of the radial flux is maintained constant. The assumption that laser flux is equal to direct heat input hðr Þ with Gaussian distribution can be described as r2

hðr Þ ¼ ce2r2

cordðNO Þ \cord ðNN Þ ¼ G 1

NC ¼ cord ðGÞ

(30) (31)

where node (NC ) denotes the common nodes between the previous and next mesh configurations f ðNC ; Eo Þ ¼ f ðNC ; EN Þ

(32)

which solves for the elements EC , common to both previous and next mesh configurations Eo  EC ¼ ENCO

This asymptotic behavior can be further exploited to increase efficiency of otherwise computationally expensive spatiotemporally periodic problems. The algorithm for achieving the increased efficiency is described in the next two subsections. A nonlinear asymptotic expansion decomposition of this problem has been attempted [33].

Fast Sorting Methodologies to Increase Data Transfer Between Meshes. The algorithm for interpolation of degree of freedom (DOF) information from a node in a previous spatial mesh configuration to the next spatial mesh configuration depends on the implemented method. In the present work, an inverse isoparametric interpolation strategy has been used. This strategy searches a target element in the next spatial mesh configuration in close proximity to the nodes present in the previous spatial mesh configuration. This search is expensive to perform though most of the nodes in the coarse mesh have a one-to-one correspondence between the present spatial mesh configuration and the next one. Henceforth, the number of calls for searching a target element in the next mesh configuration is largely reduced. Moreover, the search is not exhaustive in nature. The element in the next mesh configuration has defined bounds and the nodes falling between those bounds are used for interpolation, as shown in the schematic in Fig. 6. X and y bounds are shown with local isoparametric target element axes f and e. The fast sorting strategy is described using set theory based mathematical equations from Eqs. (28)–(37)

(33)

where element (ENCO ) denotes the uncommon nodes present in the previous mesh configuration NN  NC ¼ NNCN

(34)

where node (NNCN ) denotes the uncommon nodes present in the next mesh configuration BoundðENCO Þ ¼ ½Xini;NCO ; Xini;NCO ; Yini;NCO ; Yini;NCO ; Zini;NCO ; Zini;NCO  8cordðNNCN Þ 2 BoundðENCO Þ (35) as shown in Fig. 7. There exists a pair states that PairðNCN; NCOÞ ¼ ½NNCN ; TðNNCN Þ

(36)

(28)

where subscript O denotes the previous (old) mesh configuration and subscript N denotes the next (new) mesh configuration. Figure 6 shows nodes in two different consecutive configurations. Mesh denotes the connectivity of node (N) and element (E), respectively. Each component of the mesh array (Mesh) consists of one 041001-6 / Vol. 137, AUGUST 2015

where the cordð N Þ function denotes the three-dimensional location of a particular node (N) consisting of a position vector triplet ½XN ; YN ; ZN T

(26)

Substituting Eq. (26) into Eq. (25) and using an order of magnitude analysis along with Eqs. (7) and (10) results in Eq. (27)  2   

  r2 d T 1 # k 2 ¼ # 2  # ce2r2  # h2 (27) dr r

Mesho ¼ f ðNo ; Eo Þ MeshN ¼ f ðNN ; EN Þ

(29)

Fig. 6 Schematic showing old mesh configuration element boundaries (lines) and next mesh configuration nodes (square dots)

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Fig. 7 Target element (old mesh configuration) for illustrated nodes (next mesh configuration)

matrix is a function of local temperature dependent material properties in the coarse mesh

where TðNNCN Þ  ENCO

(37)

FFD-AMRD Thermal FEM Algorithm. In the FFD-AMRD algorithm, the refinement zone is expected to refine and derefine itself as the external spatiotemporal stimulus or response moves. Table 1 explains the intelligent algorithm to efficiently perform such refinement and derefinement in negligible time overhead for FEM mesh regeneration and renumbering along with recalculation of FEM matrices. The center of fine mesh zone has been moved based on nearest node in the original uniform coarse grid from the laser beam spot center. Efficient Stiffness Matrix Assembly. The stiffness matrix of the dynamic moving mesh for a particular offset’s mesh configuration is generated from the intact coarse mesh, negative stiffness of cut mesh and fine mesh stiffness matrices. This reduces the computational time required to assemble stiffness matrices. In this section, standard mathematical notations are used to describe submatrices. The a and b notations are index sets for specific rows and columns of A (a typical finite element stiffness matrix).

Kvac ½a; b ¼ Vac constantðf ða; bÞÞ

(38)

Kvac ða; bÞ ¼ 0

(39)

where n o mesh @ a 2 ½meshcut1 \mesh cut2 \cut3 \ … :meshcut m \ n o mesh @ … :mesh m b 2 ½meshcut1 \mesh \ \ \ cut cut2 cut3 where ½meshcuti is a connectivity vector for the ith element in the mesh that is needed to be cut from the coarse mesh. Similarly, Kfine is defined with following definition of ðp; qÞ

Aða; bÞ : submatrix of A; obtained by choosing the elements of A which do not lie in rows a and columns b A½a; b : submatrix of A ðn  nÞ; obtained by choosing the elements of A which lie in rows a and columns b The stiffness matrix of an intact coarse mesh is denoted by Kintact . The first step in stiffness matrix generation is deducing components of stiffness that will be lost due to cutting of the coarse mesh. This component will be denoted as Kvac . The elements of Kvac will be subjected to rigid translation when the finemesh configuration is moved to the location (offset þ 1). The Vac constant Journal of Manufacturing Science and Engineering

Fig. 8 3D extruded fixed coarse mesh generation

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ðcoarse nodes  cut nodesÞ < p < ðcoarse nodes  cut nodes þ fine mesh nodesÞ ðcoarse nodes  cut nodesÞ < q < ðcoarse nodes  cut nodes þ fine mesh nodesÞ Kfine ½p; q ¼ Fine constant Kfine ðp; qÞ ¼ 0

(40)

The final assembly of stiffness for a particular configuration of mesh is as follows: Kglobal ¼ PermðKintact  Kvac ÞPermT þ Kfine

Fig. 9 3D extruded independent fine mesh generation

(41)

The Perm is a row and column permutation matrix operator that assembles pieces of fragmented stiffness matrices together.

Fig. 10 FFD-AMRD mesh, stiffness, and specific heat generation matrix

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Comparison of FFD-AMRD Algorithm With Traditional FEM. Figure 10 [] shows that fine mesh box divisions (Fd) of the course mesh are transitioned in to same number of divisions. The zoom in ratio in both x and y directions (Zr) will determine the computational complexity when compared with uniform mesh FEM analysis. The computational complexity [55] of Choesky decomposition for solving (n) number of simulation equations is ðn3 =2Þ. This implies that the time complexity comparison of a uniform mesh compared with FFD-AMRD will be exactly Zr 6 . This is based on the fact that the ratio of number of nodes will be equal to ðZr 2 Þ. In the present work, since Zr equal to 3 is used, the computational complexity ratio between traditional FEM and FFD-AMRD will be 729.

Problem Geometry Description and Boundary Conditions Two case studies were attempted. The first is a linear onedimensional FEM analysis approximation of spatiotemporally periodic heat input and is referred to as case study 1. This analysis will serve as a simplified case for developing a phenomenological understanding of spatiotemporal nature, various nonlinearities and for testing algorithms. The second case study involves a threedimensional FEM simulation using the FFD-AMRD algorithm to simulate metal laser sintering and is referred to as case study 2. In both case studies, heat input (machine process) parameters are assumed as follows: scan speed : 1200 mm=s; beam diameter : 100 lm hatch spacing : 100 lm Laser power : 180 W

qðx; tÞ ¼ k

@T ðx; tÞ @x

(43)

where qðx; tÞ is the flux at point x and time t. Further, the spatiotemporal periodicity of the flux boundary is considered as qðx þ vðdtÞ; t þ dtÞ ¼ qðx; tÞ

(44)

The Dirichlet boundary condition considered here is constant temperature at the left end of the bar, as shown in Fig. 11. The Dirichlet boundary condition is required so that the thermal stiffness matrices, namely, the conduction and specific heat matrices, become nonsingular. T ð0; tÞ ¼ T0 ¼ 353 K The time-periodic Neumann boundary condition considered here is the laser flux distribution in one-dimension !   2P 2ðx  vtÞ2 exp (45) qðx; tÞ ¼ 2 2 prlaser rlaser where P is the laser beam power ¼ 180 W, rlaser is the laser beam spot size incident perpendicular to the length (l ¼ 100 mm) ¼ 100 lm, and vt is the distance travelled by the laser beam from the left end of the bar with a speed (v ¼ 1200 mm=sÞ at time instant t The initial condition for the bar considered here is constant temperature T0 T ðx; 0Þ ¼ T0 ¼ 353 K

Case Study 1. The problem geometry considered here is a onedimensional bar. The assumption of a one-dimensional bar is valid since the geometry considered is a straight bar with small crosssectional dimensions compared to its length (l). Figure 11 shows the problem geometry. The basic equation of heat transfer is 2

@T ðx; tÞ k @ T ðx; tÞ ¼ þ Qðx; tÞ @t cq @x2

(42)

where Tðx; tÞ is the temperature at point x and time t, k is the onedimensional material conductivity, c is the one-dimensional specific heat of the material, q is the density of the bar for a unit cross section, and Qðx; tÞ is the inner heat generation at point x and time t. Also, we define another variable to incorporate the flux boundary conditions as follows:

Fig. 11 Dynamic boundary condition of the one-dimensional problem

Journal of Manufacturing Science and Engineering

Material Properties. The material considered in the present work is Ti6Al4V alloy. The required Ti6Al4V properties have been provided [33]. The nonlinear temperature dependent thermal properties are demonstrated in Fig. 12. FEM Discretization and FEM Formulation. The bar is discretized into p þ 1 ¼ 2000 nodes as shown in the Fig. 13. All elements are of the same length so element matrices such as conduction and specific heat matrices involved in FEM will be the same for all elements.

Fig. 12 Plot of temperature dependent conductivity and volumetric heat capacity in SI units plotted against temperature in Kelvins used in case study 1. For Ti6Al4V material.

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Fig. 13 One-dimensional line elements

A typical C matrix of an element is as follows:  a coeffc a C¼ coeffc a a A typical K matrix of an element is as follows  b coeffk b K¼ coeffk b b where a is the specific heat capacity of a node when increasing its temperature at a rate of 1 C/s and coeffc is the interaction specific heat capacity between the connecting nodes. Similarly, b is the amount of flux flowing through a node when increasing its temperature gradient at a rate of 1 C/m and coeffk is the interaction flux between the connecting nodes. coeffc and coeffk are generally considered 0.5 for the one-dimensional case since the probability of heat flow is equal on both sides of the node. Since a onedimensional approximation of the three-dimensional heat transfer mechanisms of the metal laser sintering process are considered in the one-dimensional bar problem, the abovementioned coefficients will show drastic reduction in their values. In the current problem, the coefficients have been reduced by homogenizing a probability distribution such that it allows more heat transfer in the bulk as compared to the surface.

provides description of boundary conditions and material properties used in this case study. Governing Equations and Boundary Conditions. The governing equations are same as that of given in Governing Equations and Finite Element Formulation in FFD-AMRD. The threedimensional counterpart of flux is modified as qðr; tÞ ¼ ~ kð$T ðr; tÞÞ

where qðr; tÞ is the flux at position vector r and time t. The spatiotemporal periodicity of the flux boundary condition in three dimensions is modified as qðr þ vðdtÞ; t þ dtÞ ¼ qðr; tÞ

(47)

The Dirichlet boundary condition considered here is constant temperature at the bottom surface of the base plate as shown in Fig. 14 T ð0; tÞ ¼ T0 ¼ 353 K The time-periodic Neumann boundary condition considered here is the laser flux distribution in one dimension 



q rjr:ez ¼zmax ; t ¼ Case Study 2. A three-dimensional linear thermal case has been considered to study the FFD-AMRD algorithm. This section

(46)



!  2 ðr  vt  r0 Þjr:ez ¼zmax 2P exp 2 2 prlaser rlaser (48)

Fig. 14 Surface boundary conditions for the metal laser sintering problem which includes convection, laser flux and fixed temperature boundary conditions

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Fig. 15 Temperature distribution in 1D space

where P is the laser beam power ¼ 180 W, rlaser is the laser beam spot size incident perpendicular to the length (l ¼ 100 mm) ¼ 100 lm, vt is the displacement by the laser beam from the left end of the bar with a speed (v ¼ 1200 mm=sÞ at time instant t, r0 is the initial position vector of the laser spot on the exposed powder surface, jr:ez ¼zmax is the condition for the laser flux to always hit the top surface of the powder bed, and jj is the second norm of the included vector. The initial condition for the bar considered here is constant temperature T0 T ðr; 0Þ ¼ T0 ¼ 353 K Material Properties. The thermal conductivity ðkÞ and heat capacity (qc) has been assumed to be as follows: k ¼ 6:7 W=mK qc ¼ 2:5  106 J=ðm3  KÞ Space and Time Discretization Powder bed size : ð2  103 mÞ  ð2  106 mÞ Number of elements in x or y direction coarse mesh divisions ¼ 35 Number of coarse mesh divisions radiating into fine mesh ¼ 8 # of mesh divisions in the radial direction transitioning from coarse to fine  fine region ¼ 8 Time step size ¼ 2  106 s

Results and Discussion Case Study 1. Two subcases were attempted with constant and temperature dependent thermal parameters. They have been described as follows: Subcase 1: thermal material parameters are assumed to be constant Subcase 2: nonlinear temperature dependent thermal material parameters [33] are considered. Temperature Distribution. Figure 15 shows the temperature distribution of the one-dimensional bar considered here for both the subcases described above and at three equidistant time instances of 0.0025 s, 0.005 s, and 0.0075 s. All three time instances are plotted when melt pool distribution has achieved its steady state over time. Temperatures in all cases are normalized by dividing by the maximum temperature in the linear case over time. The rationale for doing this normalization is to make results more generalized and meaningful to compare the response between the linear and nonlinear simulations. It can be observed from Fig. 15 that spatial temperature gradients away from the laser spot are almost flat and near the room temperature compared to the near laser spot region. The near laser temperature response also demonstrates a spatiotemporal nature except at the start of the laser scan. This observation qualifies the use of FFD-AMRD algorithm involving a moving refinement zone. It can also be observed from Fig. 15 that the temperatures when using nonlinear parameters are significantly lower than when using linear parameters. The reason for this behavior can be attributed to increased conductivity and volumetric heat capacity near the laser beam spot as shown in Figs. 16 and 17, which leads to

Fig. 16 Thermal conductivity distribution for nonlinear and linear cases of the one-dimensional problem at three different times plotted against nodes

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Fig. 17 Volumetric heat capacity (qc) for nonlinear and linear cases of the one-dimensional problem at three different times plotted against nodes

Fig. 18 Comparison of thermal contours. The melt pool diameter in (b) is 125 lm.

faster heat dissipation near the melt pool area. Further the melt pool diameter is comparatively smaller in subcase 2. Thermal Conductivity. The thermal conductivity remains constant as assumed over time and space in subcase 1 whereas in subcase 2, it shows a top-hat distribution due to constant thermal conductivity at temperatures higher than 1923 K (the melting temperature) as shown in Fig. 16. Volumetric Heat Capacity. The nonlinear phenomenon in Material Properties has led to a reverse top hat distribution near the melt pool as shown in Fig. 17. The nonlinear behavior demonstrated here provides an opportunity to restrict nonlinear simulations in the near laser region and reduce computational cost significantly.

difference between the simulation and analysis can be attributed to the simplifying linearity assumption made in the analysis.

Novelty and Impact of the Present Work A novel efficient stiffness matrix assembly algorithm has been developed. It has the potential to improve traditional adaptive meshing algorithms by restricting the stiffness matrix assembly only to the area where the mesh has changed from the last mesh configuration using a stiffness matrix cut and paste approach, as presented in the present work.

Three-Dimensional FEM Description of Spatiotemporally Periodic Metal Laser Sintering Problem With Ti6Al4V Material. A three-dimensional simulation using the FFD-AMRD algorithm is performed for case study 2. Figure 18 shows the surface temperature contours for the stabilized melt pool. It can be seen that melt pool diameter is 125 lm. Microstructural Validation. Microstructural samples were created using a commercial EOS M270 metal laser sintering machine. These samples were fabricated using the same set of parameters used in the simulation and then cut to illuminate the transverse section. A set of cylindrical samples fabricated horizontally in the xy-plane of the machine (normal direction þx) were made. It can be seen in Fig. 19 that the melt pool diameter, illustrated by the grain boundary size, is around 100 lm. The 041001-12 / Vol. 137, AUGUST 2015

Fig. 19 Optical microscopy image of microstructure showing grain boundaries along the melt pool boundary

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Table 1 FFD-AMRD algorithm Step 1: Mesh Generation:

1. Fixed coarse mesh generation a  b  c size prism  divx , divy and divz ; 2D element connectivity (Matrix: # of Coarse Elements ðCoarse el lengthÞ  8)  nodex , nodey and nodez  2D mesh for bottom surface 2D connectivity (elem) and coordinate (coord) arrays  Extrude to make 3D mesh 3D elem and coord arrays (Fig. 8) 2. Dynamic fine mesh block Sd  Sd size prism  Decide mesh divisions  Generate bottom surface mesh: i. nodes for the central square box ii. nodes for radial boxes iii. element connectivity for each block  Extrude in z direction  elem and coord (Fig. 9) 3. Combined Adaptive Mesh for laser beam at location r ¼ rq þ vt  Cut elements cut list(Vector: Evac  1) from coarse mesh  Paste finemesh  GlobalConnectivity (Matrix: Numelem  8) array (Fig. 10)  GlobalCoord (Matrix: tlnodes 3) array (Fig. 10) 4. Scan Pattern Generation  time t i  key turning points OR Trajectory Points Kturni ðxi ; yi Þ  Time ! step dt i  dxi ¼vdt i  Discretization between all Kturni and Kturniþ1  Final array: Laser loc (Matrix: (tstepsþ1)3)  Laser locði; :Þ ¼ ½t i xi yi ] Step 2: Finite Element Simulation with Feed Forward Dynamic Adaptive Meshing FOR layer ¼ 1 to # layers ðNumbers of layersÞ Modify: elem and coord for added layer Decide: dimensions of all arrays like K and C Generate Kcoarse ; Kfine; Ccoarse ; Cfine Generate Kvac and Cvac FOR offset ¼ 1 to #finemesh movements from last mesh to new mesh: isoparametric interpolation

i. DOF Transfer ii. State variable FOR timestep: ti totiþn Generate Nonlinear K and C matrices Kpred ¼ f ðKcoarse Þ þ A½Kfinenorm B þ gðT; state variablesÞ Simplify in the form of Kpred ¼ K þ DK Tðiþ1Þpred ¼ choleskyðKpred ÞTi Kcorrected ¼ f ðKcoarse Þ þ A½Kfinenormcorrected B þ gðT; state variablesÞ Tðiþ1Þcorrected ¼ f 0 ðKcorrected Þ Interpolation to simple regular base grid: For input to DDCP FEM DOF Transfer State Variable Transfer END timesteps END offsets FOR 1 to # cooling time steps Generate Nonlinear K and C matrices for the fine mesh region Kpred ¼ f ðKcoarse Þ þ A½Kfinenorm B þ gðT; state variablesÞ Simplify in the form of Kpred ¼ K þ DK Tðiþ1Þpred ¼ choleskyðKpred ÞTi Kcorrected ¼ f ðKcoarse Þ þ A½Kfinenormcorrectd B þ gðT; state variablesÞ Tðiþ1Þcorrected ¼ f 0 ðKcorrected Þ END time steps END Layers

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The presented formulation offers a mathematical framework for combining adaptive meshing and moving mesh algorithms. It can use the upper bound of the mesh refinements derived from the moving mesh method and move it as a fine mesh in a coarse grid (which is synonymous to AMR and derefinement). This offers a novel discrete numerical approach for simulating continuously moving spatiotemporally periodic problems like moving energy inputs or wave propagations. An analogous continuum formulation available in traditional solid mechanics literature is the influence lines method [56] used in analysis of moving boundary conditions in solid mechanics. Future research in this paradigm will include exploiting the developed framework with the addition and combination of different numerical techniques as follows: • •

•

•

•

Asynchronous time integration for the fine and coarse scales, involving smaller time steps for the fine scale region. Reduced degree of freedom model for the fine mesh in the case of near static problems or slow moving coarse scale problems. Limiting nonlinear phenomena to the fine scale can help reduce computational cost for problems where nonlinearity is localized. Multirank updates for simultaneous equations involving change in the stiffness matrix locally in the fine mesh requiring significantly smaller time steps. Node numbering sequences involving fine mesh nodes numbered at the end will help with Cholesky decomposition multirank updates. Development of new quasi-static formulations and algorithms for moving waves within a finite element framework.

Conclusion The numerical technique developed has shown significant potential for efficient and high resolution simulation of spatiotemporal problems with rapid time and spatial gradients. The magnitude of the temporal temperature gradients is on the order of 105 C/s. Since the temperature profile is approximately Gaussian near the melt pool and is moving at the speed of 1200 mm/s traditional FEM analysis is found be 100 times slower compared to the FFD-AMRD algorithm. It is seen in the line problem approximation that the linear approximation gives higher temperatures, as well as melt pool diameters. This is also observed in 3D FFDAMRD simulations and has been validated with optical micrographs. Further development of nonlinear FFD-AMRD will help give correct temperature distributions during processing.

Acknowledgment The authors would like to acknowledge the Air Force Research Laboratory (AFRL) for support through SBIR Contract # FA8650-12-M-5153 provided to Mound Laser and Photonics Center (MLPC). The formulations, modeling and validation activities involved in this work have been made possible through a subcontract awarded to the University of Louisville (UL) by MLPC. The authors would like to thank Dr. Larry Dosser, Mr. Kevin Hartke, Mr. Ron Jacobsen from MLPC, and Mr. Tim Gornet from Rapid Prototyping Laboratory (RPC) at UL for enabling required experimental facilities during this work.

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