Peridynamic framework to model additive manufacturing processes

This paper introduces a novel framework for analyzing additive manufacturing processes. By employing a peridynamic mesh‐free implementation, it overcomes certain limitations encountered with finite element‐based approaches in regard to defects and fractures. The framework includes the implementation of thermodynamical material behavior and a novel boundary detection algorithm, as the outer surface area will undergo changes during the manufacturing process. The proposed methods are thoroughly validated, and a comprehensive example is provided as a demonstration.

F I G U R E 1 Deformation and interaction of material points  and  ′ in the non-ordinary state-based theory and their respective neighborhoods   and   ′ .
between the positions  and  ′ is defined as the integral balance of momentum ∫  ((, )⟨ ′ − ⟩ − ( ′ , )⟨ −  ′ ⟩)  +  =  ü. ( Three variations of the peridynamic model are currently being used, the bond-based, the ordinary state-based and the non-ordinary state-based formulation.In this order, flexibility increases, but so does the complexity of the formulations.
A special formulation of the non-ordinary state-based model was introduced by Silling et al. [3] in 2007.This so-called "correspondence" formulation defines an integral non-local deformation gradient to calculate the bond force vector density states, where is the shape tensor with   defined as the neighborhood volume.For each bond  =  ′ −  an influence function ⟨ ⟩, an undeformed vector state ⟨ ⟩ and a deformed vector state ⟨ ⟩ is defined.The shape tensor has to be positive definite and symmetric.The advantage of using the non-local deformation gradient is, that classical continuum mechanical models can be used in Peridynamics.The peridynamic force density vector state is thus The Piola-Kirchhoff stress tensor with respect to an orthonormal basis can be determined as (5)

Thermal flux
The mechanical response due to temperature is included in the peridynamic model.However, the heat flux must be included as well.The following derivation is based on refs.[8,9].Under the assumption that mechanical deformations do not change the temperature, the thermodynamical equilibrium equation can be studied separately to Equation (1).
In contrast to Equation (1), which is a second-order differential equation, this equation is first-order, simplifying the numerical solving process.The parameters include  the mass density,   the specific heat capacity, τ the temperature gradient in time,   the volume and   the heat sink or heat source.The heat flux of a bond is defined as with  as classical heat flux and  as the shape tensor defined in Equation (3).It follows which can be derived utilizing the spatial gradient of the temperature ∇ as is the 3 × 3 matrix of thermal conductivity, typically it is a diagonal matrix.Following [8] the spatial temperature gradient ∇ can be derived as The numerical solving process is then

Heat transfer to environment
Following [10,11] the heat volumetric density at the surface for an assigned heat flux normal to the surface   is where Δ can be set to .Thereby,   is where  is the heat convection coefficient between solid and environment, and   the environmental temperature.For a mesh-free model, the question arises: how the outer surface and the corresponded surface can be identified?For the outer surface the peridynamic neighborhood  is utilized.It is assumed that is circle for 2D and a sphere for 3D.Therefore, the following criteria has to be fulfilled for 2D and 3D Each point, which is next to the surface will have less volume represented than the discrete material points.Defining a limit value allows an easy identification of surface nodes  during the printing process.Combining Equations ( 12) and ( 13) allows the calculation of change in temperature for these nodes  as The minimum time step for the explicit time integration of the temperature field, to obtain a stable solution is given by with  the number of neighbors of point  [12].

Additive model
The additive model is realized by including a print path.This path gives each node a time stamp   .This time represents the position of the printer during the printing process.In order to activate bonds the influence function ⟨ ⟩ defined in Equations ( 4) and ( 10) is used.In the simplest case, presented here in the paper, ⟨ ⟩ is defined as The printing temperature is included, utilizing the heat source   introduced in Equation ( 6).It must be noted that after activation in Equation (20) the bonds are not in a constant state.They can be deactivated due to damages as well.
Numerically, this includes the bonds not being checked after activation.

Conclusion
The verification shows that all models were implemented in a correct way.The errors compared to the reference solutions are low and can be reduced by using finer discretizations, if the spacial distribution is not sufficient (see Figures 2 and 3).

PRINTING
This section describes how the peridynamic mesh is generated based on the G-code and the corresponding CAD (computer-aided design) model.

Printing process
To simulate the thermomechanical behavior of an additive material using Peridynamics, a geometric model is required.This entails a list of points, their corresponding volumes, and material properties for the mesh-free approximation.The process begins with the CAD model of the structure intended for printing.Typically, the printing process is controlled by the widely used G-code, a computer numerical control programming language.This code is derived from the CAD model using commercial software, and it describes the path and speed of the printing process.It is necessary to interpret and translate this numerical process information into the Peridynamic representation.
The following steps are involved in generating a discretized mesh: • CAD to Peridynamic mesh conversion The first step involves discretizing the CAD model into a mesh of volume elements.The mesh-free material pointbased Peridynamic mesh uses the center of gravity of each element as the position of the material point, with the element volume corresponding to the material point volume.For additive manufacturing, it is crucial to determine when regions become active.
Information regarding the printing path and processing speed is obtained from the G-code to determine the activation time and material orientation for the material points.Basic parameters such as nozzle width and layer height are also extracted from the G-code.
The printer's path, as described by the G-code, is extracted.The nozzle movement is mainly described by linear interpolations between fixed start and endpoints, with the corresponding velocity determined by the feed rate.Following the G-code path, the Peridynamic mesh is obtained.The maximum step width is defined by the minimum distance between two material points in the Peridynamic mesh.Since the path is represented as a line, a bounding box is placed around the print path points to identify the material points that need to be activated.These points are assigned the corresponding printing time as the virtual printer reaches them.As depicted in Figure 4, every Peridynamic node intersected by the bounding box receives the appropriate printing time and orientation.This approach allows for arbitrary discretizations, as multiple points can be activated simultaneously.
If the assigned time-value of the nodes is reached by the peridynamic simulation time, the node and its neighborhood will be activated.Hence, we are able to simulate the temperature distribution of additive materials.

CONCLUSION
This paper demonstrates the implementation and verification of thermodynamical material behavior, it also introduces a novel dynamic surface detection algorithm.By combining this algorithm with the printing time information from the G-code, we showcase the peridynamic framework's ability to simulate additive manufacturing processes.As a concrete example, Figure 5 illustrates the final temperature of a newly printed L-Angle probe.Due to computational time, only the temperature distribution was calculated.For further research, the full thermoelastic material behavior will need to be analyzed.Moreover, material effects that will occur, such as phase transitions, need to be implemented.This could be achieved with an existing User Material (UMAT) subroutine.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

F I G U R E 2 F I G U R E 3
Geometry and discrete model for the verification of the cooling process based on heat flow.Geometry and discrete model for the verification of the cooling process based on heat convection.
Extracting parameters from G-code • Retrieving printing path from G-code • Calculating node timing and orientation • Identifying Peridynamic nodes within a bounding box along the printing path • Assigning timing and orientation to mesh points • Writing the converted mesh F I G U R E 4 Conversion schema.F I G U R E 5 Peridynamic L-angle simulation results.