Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen-
Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen- * Corresponding author.
E-mail address: sxb514@psu.edu (Saurabh Basu). tered cubic unit cell with vertical and horizontal struts (FBC-CXYZ) [10, 11] . Interest in lattice structures for various applications arises primarily from their superior strength to weight ratio. This characteristic may also simultaneously benefit extraneous functions, e.g. ventilation assisted mitigation of corrosion [12, 13] and acoustic properties. Research on mechanical behavior of lattice structures has looked at their mechanical properties and characteristics under different load conditions such as compression, shear or blast loading [14, 15, 16, 17, 18] .
Unfortunately, AM-fabricated parts normally suffer from poor surface roughness and porosity defects. These imperfections naturally arise during the AM process [19, 20] and adversely affect resulting mechanical, e.g. elastic [21] and plastic [22] deformation behavior. Hence, these defects must be mitigated to achieve ideal performance from lattice structures. However, efficacy of approaches for their mitigation, e.g. postprocessing and hot iso-static pressing decreases as the complexity of lattice structure geometry increases. In this regard, recent research efforts have looked at approaches to encapsulate effects arising out of tolerable defects in the design of lattice structures. From the context of finishing lattice structures to make their defects tolerable, the question naturally arises, what is tolerable or good enough.
Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen- * Corresponding author.
E-mail address: sxb514@psu.edu (Saurabh Basu). tered cubic unit cell with vertical and horizontal struts (FBC-CXYZ) [10, 11] . Interest in lattice structures for various applications arises primarily from their superior strength to weight ratio. This characteristic may also simultaneously benefit extraneous functions, e.g. ventilation assisted mitigation of corrosion [12, 13] and acoustic properties. Research on mechanical behavior of lattice structures has looked at their mechanical properties and characteristics under different load conditions such as compression, shear or blast loading [14, 15, 16, 17, 18] .
Unfortunately, AM-fabricated parts normally suffer from poor surface roughness and porosity defects. These imperfections naturally arise during the AM process [19, 20] and adversely affect resulting mechanical, e.g. elastic [21] and plastic [22] deformation behavior. Hence, these defects must be mitigated to achieve ideal performance from lattice structures. However, efficacy of approaches for their mitigation, e.g. postprocessing and hot iso-static pressing decreases as the complexity of lattice structure geometry increases. In this regard, recent research efforts have looked at approaches to encapsulate effects arising out of tolerable defects in the design of lattice structures. From the context of finishing lattice structures to make their defects tolerable, the question naturally arises, what is tolerable or good enough.
Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen- * Corresponding author.
E-mail address: sxb514@psu.edu (Saurabh Basu). tered cubic unit cell with vertical and horizontal struts (FBC-CXYZ) [10, 11] . Interest in lattice structures for various applications arises primarily from their superior strength to weight ratio. This characteristic may also simultaneously benefit extraneous functions, e.g. ventilation assisted mitigation of corrosion [12, 13] and acoustic properties. Research on mechanical behavior of lattice structures has looked at their mechanical properties and characteristics under different load conditions such as compression, shear or blast loading [14, 15, 16, 17, 18] .
Unfortunately, AM-fabricated parts normally suffer from poor surface roughness and porosity defects. These imperfections naturally arise during the AM process [19, 20] and adversely affect resulting mechanical, e.g. elastic [21] and plastic [22] deformation behavior. Hence, these defects must be mitigated to achieve ideal performance from lattice structures. However, efficacy of approaches for their mitigation, e.g. postprocessing and hot iso-static pressing decreases as the complexity of lattice structure geometry increases. In this regard, recent research efforts have looked at approaches to encapsulate effects arising out of tolerable defects in the design of lattice structures. From the context of finishing lattice structures to make their defects tolerable, the question naturally arises, what is tolerable or good enough.
Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen- * Corresponding author.
E-mail address: sxb514@psu.edu (Saurabh Basu). tered cubic unit cell with vertical and horizontal struts (FBC-CXYZ) [10, 11] . Interest in lattice structures for various applications arises primarily from their superior strength to weight ratio. This characteristic may also simultaneously benefit extraneous functions, e.g. ventilation assisted mitigation of corrosion [12, 13] and acoustic properties. Research on mechanical behavior of lattice structures has looked at their mechanical properties and characteristics under different load conditions such as compression, shear or blast loading [14, 15, 16, 17, 18] .
Unfortunately, AM-fabricated parts normally suffer from poor surface roughness and porosity defects. These imperfections naturally arise during the AM process [19, 20] and adversely affect resulting mechanical, e.g. elastic [21] and plastic [22] deformation behavior. Hence, these defects must be mitigated to achieve ideal performance from lattice structures. However, efficacy of approaches for their mitigation, e.g. postprocessing and hot iso-static pressing decreases as the complexity of lattice structure geometry increases. In this regard, recent research efforts have looked at approaches to encapsulate effects arising out of tolerable defects in the design of lattice structures. From the context of finishing lattice structures to make their defects tolerable, the question naturally arises, what is tolerable or good enough.
Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen- * Corresponding author.
E-mail address: sxb514@psu.edu (Saurabh Basu). tered cubic unit cell with vertical and horizontal struts (FBC-CXYZ) [10, 11] . Interest in lattice structures for various applications arises primarily from their superior strength to weight ratio. This characteristic may also simultaneously benefit extraneous functions, e.g. ventilation assisted mitigation of corrosion [12, 13] and acoustic properties. Research on mechanical behavior of lattice structures has looked at their mechanical properties and characteristics under different load conditions such as compression, shear or blast loading [14, 15, 16, 17, 18] .
Unfortunately, AM-fabricated parts normally suffer from poor surface roughness and porosity defects. These imperfections naturally arise during the AM process [19, 20] and adversely affect resulting mechanical, e.g. elastic [21] and plastic [22] deformation behavior. Hence, these defects must be mitigated to achieve ideal performance from lattice structures. However, efficacy of approaches for their mitigation, e.g. postprocessing and hot iso-static pressing decreases as the complexity of lattice structure geometry increases. In this regard, recent research efforts have looked at approaches to encapsulate effects arising out of tolerable defects in the design of lattice structures. From the context of finishing lattice structures to make their defects tolerable, the question naturally arises, what is tolerable or good enough.
Additive manufacturing (AM) has become a popular materials processing method for rapidly fabricating complex tailormade geometries [1, 2] . Stainless steels, nickel and titanium alloys are common choices for fabricating parts with AM due to their utility in high-performance applications [3, 4, 5, 6, 7] . Common AM configurations for fabricating these components include methods like electron beam melting (EBM) and selective laser melting (SLM) [8, 9] .
Control over these manufacturing variables, e.g. material and process, make AM ideally suited for fabricating lattice structures for high performance applications in different sectors. These structures refer to topologically ordered and open cell lattice-like configurations such as body centered cubic unit cell with vertical struts (BCCZ), face and body centered cubic unit cell with vertical struts (FBCCZ), face and body cen- * Corresponding author.
E-mail address: sxb514@psu.edu (Saurabh Basu). tered cubic unit cell with vertical and horizontal struts (FBC-CXYZ) [10, 11] . Interest in lattice structures for various applications arises primarily from their superior strength to weight ratio. This characteristic may also simultaneously benefit extraneous functions, e.g. ventilation assisted mitigation of corrosion [12, 13] and acoustic properties. Research on mechanical behavior of lattice structures has looked at their mechanical properties and characteristics under different load conditions such as compression, shear or blast loading [14, 15, 16, 17, 18] .
Unfortunately, AM-fabricated parts normally suffer from poor surface roughness and porosity defects. These imperfections naturally arise during the AM process [19, 20] and adversely affect resulting mechanical, e.g. elastic [21] and plastic [22] deformation behavior. Hence, these defects must be mitigated to achieve ideal performance from lattice structures. However, efficacy of approaches for their mitigation, e.g. postprocessing and hot iso-static pressing decreases as the complexity of lattice structure geometry increases. In this regard, recent research efforts have looked at approaches to encapsulate effects arising out of tolerable defects in the design of lattice structures. From the context of finishing lattice structures to make their defects tolerable, the question naturally arises, what is tolerable or good enough. Computational methods provide an effective approach to predict effects of defects in lattice structures that result from AM. This methodology involves planting of defects featuring a statistically similar makeup as those observed in real lattice structures in their computational counterparts. This is followed by application of displacement or force boundary conditions to the computational structures and characterization of their response. The same methodology is also effective in simulating the material state that results directly from the AM characterized by its residual stress field [23] .
Due to high computational requirements when deformation of lattice arrays comprising a large number of repeat units is simulated, the aforementioned approach is adapted with mechanically equivalent members [24, 30, 31 ] that feature statistically similar defects while reducing the number of elements required for simulation. Although this approach accelerates the simulation process, the methodology ignores volume porosity defects that are common place in components resulting from AM, focusing primarily on effect of surface roughness. Studies that have attempted to simulate effects of porosity defects 2 In the present study, a FEM based computational framework is created to understand the effects of surface roughness and porosity on mechanical response of 2.5D (e.g. plane-stress) lattice structures. Effects of roughness and porosity area fraction are normalized with respect to strut thickness and analyzed with respect to the number of repeat units in the structure. Novel contributions of the work include delineation of methodology for creating numerical frameworks for testing of various stochastic distributions of defects in various structures. Herein, artefacts arising from pseudo-randomness of stochastic defects generated in computer during their numerical implantation are discussed. Utility of this methodology is demonstrated in parametric delineation of effects of defects on various lattices that feature different designs and defect structures. 
Two material attributes of Inconel 718 were implemented in the FEM framework:(i) Isotropic elasticity, and, (ii) Johnson Cook (JC) plasticity. The JC model is specified by Eqn.1 [27] . The material properties of Inconel 718 used in this research are listed in table 3 [25, 26, 27] . Simulations were executed in plane stress mode. Plane stress thickness was not specified and hence the unit of reaction force in the output of the FEM framework was (N/m). Three-node linear plane stress element CPS3 and four-node bilinear reduced integration plane stress element CPS4R were used. Stress fields and reaction forces resulting from the imposed displacement boundary conditions were characterized. The choice of plane stress deformation mode in the present work via 2.5D structures results in simplification of analysis and may not capture insights pertinent to the behavior of complex 3D structures. However, these simple 2.5D structures were intentionally chosen in the present exploratory analysis to eliminate variables, i.e. more complex stress states that exist in 3D structures. This was done for testing the robustness of this computational methodology in delineating effects of roughness and porosity defects in simplified, e.g. 2.5D plane stress settings. Efforts are underway to address analogous questions in 3D structures. 
In order to assess the effect of real roughness textures that originate from AM, Inconel 718 specimens were printed using direct metal laser sintering in an EOS M 280 machine. The laser spot diameter was set at 87.5 µm and the layer thickness was set at 40 µm. The laser power level was preset at 285 W. The as-received surface of the specimen was characterized by exposing it's cross section by electric discharge machining. Subsequently, the cross section was imaged using scanning electron microscopy (SEM) as seen in Fig. 2a . A MATLAB script was used to extract discrete data points from this cross section. Fourier series was used to fit data points as given in Eq. 2 [28] , 4 and shown in Fig. 2b . In this manner, a real surface roughness profiles was quantified as a numerical roughness pattern that was then processed to be implanted onto the simulated specimen using the Python interface of Abaqus. The Fourier series pattern was repeated to cover the length of the specimen.
Unfortunately, this approach of assessing effects of surface roughness is computationally expensive due to a minimum density of surface nodes required to adequately represent texture. For instance, a larger node density results in a larger number of elements in the simulation thus slowing it down. Clearly, inclusion of smaller wavelengths in the Fourier series for representation of textures dictates greater required node density by the Nyquist criterion and hence, will result in a slower simulation. This is important from the context of optimization as the same surface texture may be represented by a larger node density to greater accuracy, but this endeavor may only provide diminishing additional insights. To assess these effects, different number of points were used to discretize the same roughness function (Fig.2c-f ). Their reaction force at the final time step upon application of the boundary conditions described in section 2.1 was extracted from the FEM model and compared.
In addition to surface roughness obtained from real specimen, the effect of ideal, e.g. sinusoidal roughness on mechanics was also analyzed. Sinusoidal function y = a sin( 2π λ x) was used to generate roughness on struts, where wavelength λ is set to 1.076mm. The amplitudes a and their corresponding roughnesses R a that were tested include parameters (a, R a ) = (4 × 10 −5 , 23µm), (4 × 10 −4 , 231µm), (8 × 10 −4 , 464µm). Here, parameter R a refers to R a = 1 n n i=1 | y i | , where y is the deviation of the assessed profile. The effect of these roughnesses was assessed by imposing the same displacement boundary conditions as described in section 2.1.
The effect of random porosity defects was assessed in terms of: (i) density, (ii) size, and (iii) algorithm for generation of random number that enables implantation of random pores. These numerical specimens were generated as described in sections 2.4.1, 2.4.2 and 2.4.3 and subject to the same boundary conditions as described in section 2.1 to assess their defects. Random number seeds 4&10 were used for the left and right leaning struts unless specified otherwise.
A preset number n = 0 − 120 of porosity defects was implanted in random locations on a unit lattice structure specimens featuring a surface roughness R a = 231µm. The defects were generated as circles with radii sampled from uniform distributions whose range was set in (0.05t strut , 0.2t strut ). Subsequently, these circles were deformed and made elliptical by stretching one axis to provide them with an aspect ratio that was sampled from a uniform distribution whose range was set in (0.7, 1).
Porosity defects amounting to 2.72% area fraction were implanted in the simulated unit lattice structure specimens at random locations. The shapes of these defects were elliptical in this suite of plane stress simulations. In order to test the effect of size of porosity defects, these ellipses were generated by sampling radii of circles from uniform distribution whose ranges were set as (0.05t strut , 0.2t strut ), (0.1t strut , 0.2t strut ), (0.15t strut , 0.3t strut ), (0.2t, 0.35t strut ), where t strut is the strut thickness. Subsequently, aspect ratios were sampled from uniform distribution whose range was set between (0.7, 1) to deform these circles into ellipses. Finally, the struts were implanted with a default surface roughness R a = 231µm, where t strut is the thickness of the parent strut. In these simulations, the porosity on the top and bottom shell were also set at 2.72% by default.
Random numbers were generated for this research by function PRNG in python that needs specification of a random number seed. The supporting inbuilt algorithm is often used by other numerical computing programs that also take a seed as input for generating random numbers. This renders a pseudo randomness to the list of number generated, making them completely reproducible as long as the same seed is specified again. This implies that results of these simulations are highly influenced by the chosen seed. This effect is analyzed by generating random numbers with various seeds for creating porosity defects and assessing variations in their behavior using the same boundary conditions as described in section 2.1. In order to minimize the effect of pseudo randomness, position and dimensions of elliptical porosity defects on top and bottom shells were kept constant, wherein area fraction of defects was maintained at 2.72%. Further, the seed of PRNG was set at 1 for the top shell and 2 for the bottom shell. The instance with seed 4 for the strut inclined towards top left, and 10 for the strut inclined to top right was used as reference. 
The effect of resolution for discretization of the real surface roughness was evaluated by simulating deformation of these lattice structures using the conditions described in section 2.1. The reaction stress along with stress fields at the end of the deformation for various levels of discretizaiton are shown in Figs. 2c-2f. Figure 2c shows that a higher discretization level creates a smaller reaction stress. This effect arises from a competition between roughness and waviness. Roughness of a surface refers to prominent wave components that feature smaller wavelengths, whereas waviness generally refers to wave components with larger wavelengths. As long as the waviness of a component surface does not compromise its structural response by activating unstable deformation modes (e.g. buckling), a wavy surface results is greater stiffness compared to a rough surface. Herein, discretization of the real roughness profiles with coarser resolution naturally provides a low pass filter effect due to which, the effect of smaller wavelength roughness features are eliminated. This hypothesis was further evaluated by testing the response of structures featuring sine waves with amplitudes and roughness (a, R a ) = (4 × 10 −4 , 231µm), and wavelengths 1.1 mm < λ < 5.5 mm. The responses of these 1 × 1 structures were collected in a manner described in section 2.1. It is seen that an increase in wavelength results in a stiffer response.
Sinusoidal roughness textures featuring (a, R a ) = (4 × 10 −5 , 23µm), (4 × 10 −4 , 231µm), and (8 × 10 −4 , 464µm) with wavelength λ = 1.076 mm were implemented on lattices. A uniform displacement boundary condition was imposed as described in section 2.1. Stress-strain curves for lattice structures with different surface roughness and various number of repeat units were simulated and compared as seen in Fig. 4a . A nonlinear relation between stress and strain was seen at a compressive strain ∼ 0.8%. The stiffness of the structure was found to monotonically decrease as the magnitude of surface roughness R a increased as seen in Fig. 4b . Similarly, the stiffness also decreased with an increasing number of repeat units in the lattice. However, this effect decays as a larger number of repeat units is added to the lattice.
A preset number between n = 0 − 120 of elliptical porosity defects was implanted in unit lattices featuring a roughness R a = 23µm, prior to application of boundary conditions described in section 2.1. The stress at the strain of 1% and the resulting stiffness of lattice structures within the elastic region in terms of porosity are shown in Figs. 5a, 5b, respectively. It is seen that 2.72% porosity defects each in the top and bottom (horizontal) struts decrease the stress at 1% strain by 4 MPa with respect to the reference sample which did not have any defects (this sample had the same surface roughness and was subject to the same boundary condition as rest of the samples described in this section). In comparison, introduction of an additional 2.98% porosity defects in oblique struts resulted in a steep decline in the stress at 1% strain from 98 to 62 MPa. This is because stress on top and bottom horizontal struts are several orders lower than stress on struts and hence, play a minor role in accommodation of load. Hence, introduction of even a small number of porosity defects in oblique struts results in a steep decline in stiffness. However, the rate of increase in depreciation in mechanical response decreased significantly with increase in defect density. Figure 5b shows the effect of pore size on the resulting stiffness. Trends illustrated in the figure generally show that porosity diameter plays only a secondary role, wherein the primary effect of depreciation of stiffness of a lattice structure arises from the defect area fraction.
In order to illustrate the impact of pseudo-randomness, 1 × 1 lattice structures with surface roughness R a = 23µm, number of porosity defects n = 20, radius range of porosity defects U(0.05t strut , 0.2t strut ) and aspect ratio U(0.7, 1) were created for numerical deformation simulations. However, the random number seed was changed to various values that are shown in Fig.  6b . In this figure, the x-axis represents seed for the two oblique struts. For instance, 4&10 in Fig. 6b refers to seed value = 4 for strut that was inclining towards the left and seed 10 for strut that was inclining towards the right. This numerical sample 4&10 was set as reference because all pores that generate on struts in previous sections use this seed combination to generate random numbers. Seed number 1 and 2 were set by default for top shell and bottom shell, respectively. Seed in these cases were used to pseudo-randomly generate numbers for position, radius and aspect ratio of elliptical pores.
It is clear that changes to the random number seed can produce significant variation in the mechanical response of lattices. This is evident from the seed number combination 5&7 that results in a force response which is 12.3% greater than the reference. On the other hand, the seed number combination 18&20 results in a reaction force that is 15.7% lower than the reference.
The effect of strut angle θ on stiffness of 1 × 1 lattice structures is shown in Fig. 6b . Here, the angle θ corresponds to the oblique angle between two adjacent struts in the structure. The stiffness increases with the increasing θ due to transition from bending dominated to uniaxial compression dominated deformation at higher strut angles. For instance, stiffness values of 2.8 × 10 10 N/m 2 and 1.1 × 10 10 N/m 2 were seen in struts featuring in θ = 75 • and θ = 48 • , respectively. In addition to this obvious effect, and interesting observation was made from this effort, viz. the rate of decline in stiffness with increasing porosity is greater in slender struts featuring higher angle θ. Further, 6 the effect of roughness is also more pronounced in larger strut angles θ.
The present work creates a workflow that can be used to studying the effect of defects on mechanical response of 2.5D lattice structures. Insights obtained from this work can be used for the following efforts.
1. Establishing a pipeline to produce statistically significant mechanical response data for automating the design of lattice structures. Modern advancements in machine learning have enabled frameworks that can accelerate discovery of microstructures. These frameworks rely on training data which is often created using simulations of various combinations of microstructures and loading conditions. Towards this concept, the present work shows an approach to create efficient simulations involving lattice structures. For instance, the present work shows the effect of various res-