## 3 Thermal Effects Prediction on Shape Deformation via Infrared Thermogram  
### Infrared Camera and Digital Camera  
The 3D printing is realized by stacking between the multilayers with various and uneven temperatures along the trajectory. Discriminating visibility is the key technology to realize virtual printing. The visible primitives including points and facets are obtained based on convex hull principles. For the mathematical description of the three-dimensional heat conduction problem, it is necessary to combine the Fourier law and the principle of conservation of energy in order to analyze the multilayer thermodynamic equilibriums state. Inspired by monocular vision and even binocular vision, Figure 3 shows the schematic binocular stereo vision measurement of layered printing. Figure 2: Six kinds of infilling patterns of femur (previously shown in Figure 1) at three layers with incremental \(h_{n}\), where (**a**) is concentric with \(h_{n}=0.23\), \(L=2162.929\) and \(n_{point}=46\); (**b**) is cross with \(h_{n}=0.23\), \(L=2215.543\) and \(n_{point}=310\); (**c**) is triangle with \(h_{n}=0.42\), \(L=1437.539\) and \(n_{point}=61\); (**d**) is tri-hexagon with \(h_{n}=0.42\), \(L=1413.658\) and \(n_{point}=62\); (**e**) is line with \(h_{n}=0.73\), \(L=822.416\) and \(n_{point}=42\); (**f**) is grid with \(h_{n}=0.73\), \(L=830.254\) and \(n_{point}=29\). The melting energy consumption as follows:  
\[E_{thermal}=cm(T_{m}-T_{a})+mX=\rho V[c(T_{m}-T_{a})+X], \tag{8}\]  
where \(E_{thermal}\) is the melting energy consumption (k)); \(c\) is the material specific heat capacity (J-kg-1.K-1); \(m\) is the weight of the filament (kg); \(T_{m}\) is the material melting point (K); \(T_{a}\) is the environment temperature (K); \(X\) is the latent heat (k)-kg). The thermodynamic equilibriums equation for multilayer satisfies the following classical 3D heat conduction equation. \[\rho c\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{(}k_{x} \frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}k_{y} \frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}k_{z }\frac{\partial T}{\partial z}\bigg{)}+\dot{q}, \tag{9}\]  
where \(\rho\) is the material density (kg-\(m^{-3}\)); \(c\) is the material specific heat capacity (J-kg-1.K-1); \(T\) is the temperature (K); \(t\) is the interaction time (s); \(k_{x},k_{y},k_{z}\) are thermal conductivities \(\left(W\cdot m^{-1}\cdot K^{-1}\right)\) along the directions of \(x\), \(y\), \(z\) axes; and \(\dot{q}\) is the internal heat density (W\(\cdot m^{-3}\)). When \(k_{x},k_{y},k_{z}\) are the same value as a constant, the thermodynamic equilibriums equation for multilayer can be obtained from the basic principles of heat transfer and the first law of thermodynamics, as follows:  
\[\frac{\partial T}{\partial t}=\alpha\bigg{(}\frac{\partial^{2}T}{\partial x^{ 2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}} \bigg{)}+\frac{\dot{q}}{\rho c}, \tag{10}\]  
\[\alpha=k/(\rho c) \tag{11}\]  
where \(\alpha\) is the thermal diffusivity \(\left(m^{2}\cdot s^{-1}\right)\), \(k\) is the thermal conductivity \(\left(W\cdot m^{-1}\cdot K^{-1}\right)\); \(c\) is the material specific heat capacity (J-kg-1.K-1) as a constant. The heat conduction equation is a partial differential equation of first order for time. But it is a partial differential equation of second order for space. Thus, solving the equation requires a  
Figure 3: Schematic binocular stereo vision measurement of layered printing. time condition for temperature and two spatial conditions. These conditions are given as the initial conditions at \(\mathrm{t}=0\) and the boundary conditions of the space. The first type of boundary condition for the temperature is defined as follows:  
\[T_{x=0}=T_{s}(t). \tag{12}\]  
When the temperature \(T_{s}\) on the boundary surface is independent of time as a constant, it is called constant wall temperature condition.