The holistic simulation of transient dynamical-thermal-optical systems combines mechanical simulations, thermal system analysis and optical simulations. The method and an exemplary numerical example are described in more detail in the following.

### Mechanical modeling of dynamical-optical systems

In the mechanical part of the holistic simulation the optical system, consisting of optical elements and their holdings, is modeled as EMBS [9]. In a first step, each body is modeled separately. Optical sensitive components such as optical lenses are considered to be elastic bodies while the remaining less important elements such as the holdings are considered as being rigid. For each elastic body, a finite element model with *N* degrees of freedom (DOF) is built. The deformations of each modeled body are described by its shape functions *Ψ* and its nodal displacements vector \({\boldsymbol {q}}(t)\in \mathbb {R}^{N}\) which varies dependent on the time *t*. The equations of motion for each elastic body result from the linear FE model and read

$$ \boldsymbol{M}_{\text{ss}}\ddot{\boldsymbol{q}}(t)+\boldsymbol{D}_{\text{ss}}\dot{\boldsymbol{q}}(t)+\boldsymbol{K}_{\text{ss}}{\boldsymbol{q}}(t)=\boldsymbol{f}(t)\;. $$

(1)

Thereby, the system matrices are the mass matrix *M*_{ss}, the damping matrix *D*_{ss}, and the stiffness matrix *K*_{ss}. The system input is the vector of the exciting forces *f*. The normally large number of DOF of FE models leads to a high computational effort for the analysis of their dynamical behavior. By means of a model order reduction (MOR) method, the number of DOF can be significantly reduced after the determination of the system matrices and the force vector. In the MOR scheme, a projection matrix \(\boldsymbol {V}\in \mathbb {R}^{N\times n}\) with *n*≪*N* is calculated. This matrix enables the approximation of the nodal displacements by

$$ {\boldsymbol{q}}\approx\boldsymbol{V} \bar{{\boldsymbol{q}}}\; $$

(2)

with the reduced coordinates \(\bar {{\boldsymbol {q}}}(t)\in \mathbb {R}^{n}\). The projection matrix is applied to the system matrices and the system input vector which leads to a reduction of the whole equation system size of Eq. (1). The reduction allows for an efficient analysis of the dynamical behavior of each modeled body. The detailed procedure description for a MOR and different methods for the calculation of *V* are given in [10, 11]. The application of MOR methods on optical systems is discussed in [12, 13]. Subsequently to the MOR, the rigid and elastic bodies are assembled to the EMBS. Then, the dynamic simulation of the assembled system, i.e. a time integration, yields the time-dependent kinematics of the complete system and the deformations of the elastic bodies.

Apart from the usage in the MOR, the projection matrix is also provided to the FE model of each elastic body if stresses need to be considered. There, it is used for obtaining the deformation state of the elastic body. By the use of this deformation state and material laws the matrix of the reduced stress functions \(\bar {\boldsymbol {\Psi }}_{\mathrm {\sigma },\mathrm {P}}\in \mathbb {R}^{6\times n}\) at a discrete point *P* within the elastic body is calculated. The matrix \(\bar {\boldsymbol {\Psi }}_{\mathrm {\sigma },\mathrm {P}}\) and the reduced nodal displacements vector \(\bar {{\boldsymbol {q}}}\) allow for the calculation of the vector of the time-dependent stresses

$$ \hat{\boldsymbol\sigma}_{\mathrm{P}}=\left[\sigma_{\mathrm{x}},\sigma_{\mathrm{y}},\sigma_{\mathrm{z}},\tau_{\text{xy}},\tau_{\text{yz}},\tau_{\text{xz}}\right]_{\mathrm{P}}^{\top} = \bar{\boldsymbol\Psi}_{\mathrm{\sigma},\mathrm{P}}\bar{{\boldsymbol{q}}} $$

(3)

at the point *P*. The normal stresses *σ*_{x},*σ*_{y}, and *σ*_{z} in the direction of the reference system in *P* and the shear stresses *τ*_{xy},*τ*_{yz}, and *τ*_{xz} in the associated planes can be rearranged to the stress tensor *Σ*_{P}. By the use of an eigen analysis of *Σ*_{P} the principal stresses *σ*_{I,P},*σ*_{II,P}, and *σ*_{III,P} are computed. A more detailed explanation on the calculation of the stresses can be found in [14, 15].

In summary, the EMBS dynamical simulation allows for the computationally efficient calculation of the time-dependent deformations and principal stresses in the elastic bodies and the time-dependent rigid body motion of all bodies.

### Thermal finite element analysis

In the thermal analysis part of the holistic simulation of the dynamical-thermal-optical system, an FE model is built for all optical lenses of the system. Normally, the FE mesh for the thermal analysis is coarser than the mechanical mesh. This saves computing time and nevertheless leads to sufficiently precise results. The calculation of the time-dependent nodal temperatures *𝜗*(*t*) in each optical lens is performed using the dynamical equation for the heat conduction

$$ \boldsymbol{D}_{\text{tt}}\dot{\boldsymbol{\vartheta}}+\boldsymbol{K}_{\text{tt}}\boldsymbol{\vartheta}=\boldsymbol{\phi}\;. $$

(4)

The matrix *D*_{tt} for the specific heat, the matrix *K*_{tt} for the thermal conductivity, and the thermal input vector *ϕ* which represents the thermal heat flux at the nodes, are computed with FE methods. While thermally induced changes of the refraction index are considered in the presented method, thermally induced deformations are not considered here.

### Optical modeling of dynamical-thermal-optical systems

The dynamical and thermal effects in optical systems can be analyzed using ray tracing. Thereby, the propagation characteristics of light rays in different media and especially media with an inhomogeneous refraction index must be taken into account. The used ray tracing approach is described in the following.

Ray tracing is a part of the geometrical optics where light rays determine the imaging performance of an optical system and wave-optical effects are neglected. In homogeneous media the rays propagate along straight lines and change their direction of travel only at the surfaces. In sequential ray tracing through homogeneous media, the ray positions *r* and the ray directions *d* are calculated in a relative surface formulation from surface to surface, from the object plane to the image plane. Thereby, the calculated ray position *r*_{i} at a surface *S*_{i} is the intersection point of a ray and the surface. The corresponding ray direction *d*_{i} results from the previous ray direction *d*_{i−1}, the intersection point *r*_{i}, the surface normal *n*_{i}, the refraction index of the previous media *n*_{i−1}, and the current one *n*_{i} [16]. Light propagates as straight line through homogeneous media. An inhomogeneous refraction index within a medium influences the path of a light ray and makes it curved. The resulting bending of the ray depends on the gradient of the refraction index. In order to trace a ray through a GRIN medium, the partial differential ray equation

$$ \frac{\mathrm{d}}{\mathrm{d} s}\left(n(\boldsymbol{r})\frac{\mathrm{d} \boldsymbol{r}}{\mathrm{d} s}\right)=\nabla n(\boldsymbol{r}) $$

(5)

must be solved numerically. A derivation of this equation can be found in [17]. The path parameter *s* can be used to approximate the differential geometrical path length d*s* by \(\mathrm {d} s = \mathrm {d} \bar {s} n\). In combination with (5) one gets

$$ \frac{\mathrm{d}^{2} \boldsymbol{r}}{\mathrm{d} \bar{s}} = n(\boldsymbol{r})\nabla n(\boldsymbol{r}) $$

(6)

which can be solved using a fourth order Runge-Kutta method [15, 18]. With this method a ray can be traced efficiently through a medium with an arbitrary inhomogeneous refraction index in small spatial steps till the back surface of the inhomogeneous medium is reached. In every spatial step the changes in the refraction index are calculated and considered in the calculation of a new ray position *r* and a new ray direction *d*. The only requirement is that the refraction index distribution within the inhomogeneous medium is known during the ray tracing.

### Dynamical-thermal-optical system modeling and simulation

The analysis of transient dynamical-thermal-optical systems requires the combination of the mechanical, the thermal, and the optical system analysis. Therefore, the results of the dynamical EMBS simulation and the results of the thermal FE analysis must be transformed in such a way that the results can be considered in the ray tracing. From the EMBS simulation we get the transient kinematic behavior of the system. Thereby, the motion is described relative to an inertial system frame. For the sequential ray tracing, all rigid body motions and elastic deformations must be transferred in a relative surface description for each time step and each surface. The time-dependent deformations of the surfaces must additionally be transformed in a way that the surfaces are described smooth enough to avoid discretization effects during the calculation of intersection points of the rays and the surfaces. Hence, the deformations must be approximated using polynomial functions and in this case we use Zernike polynomials. These polynomials can be evaluated very efficiently during the ray tracing and they can reproduce random deformation states of circular lenses. Deformations of free-form surfaces can be approximated smoothly using B-splines. The third result from the mechanical simulation, are the time-dependent principal stresses which lead to a spatial variation of the refraction index within deformed optical elements. The change of the refraction index *Δ**n*_{σ,j} due to stress at a position *r*_{j} in the medium or on the surface of an optical medium result from the stress optic law

$$ \Delta n_{\mathrm{\sigma},j}=C\left(\sigma_{\mathrm{I},j}+\sigma_{\text{II},j}+\sigma_{\text{III},j}\right) $$

(7)

with the material dependent stress-optical coefficient *C*. Thereby, polarization effects are neglected. The stress dependent refraction index changes are approximated using cylindrical polynomials, to avoid discretization effects. Therby, Zernike polynomials *Z*(*x*,*y*) of the order \([0,\dots,n_{\mathrm {Z}}]\) are used as the radial component of the cylindical polynomials and Chebychev polynomials *T*(*z*) of the order \([0,\dots,n_{\mathrm {T}}]\) describe the height component. The axis of symmetry of the cylinder corresponds with the z-axis of the back surface coordinate system of the lens. The height and the radius are chosen so that all nodes of the deformed lens are inside the cylinder. The refraction index changes at a random point **p**=(*x*_{p},*y*_{p},*z*_{p}) within the lens can be approximated using

$$ \Delta n_{\mathrm{\sigma}}(x_{\mathrm{p}},y_{\mathrm{p}},z_{\mathrm{p}}) \approx \left[\begin{array}{c} Z_{0}(x_{\mathrm{p}},y_{\mathrm{p}})T_{0}(z_{\mathrm{p}}) \\ Z_{0}(x_{\mathrm{p}},y_{\mathrm{p}})T_{1}(z_{\mathrm{p}})\\ \vdots\\ Z_{n_{\mathrm{Z}}-1}(x_{\mathrm{p}},y_{\mathrm{p}})T_{n_{\mathrm{T}}}(z_{\mathrm{p}})\\ Z_{n_{\mathrm{Z}}}(x_{\mathrm{p}},y_{\mathrm{p}})T_{n_{\mathrm{T}}}(z_{\mathrm{p}}) \end{array}\right]^{\top} \cdot \mathbf{c}\; $$

(8)

with the cylindrical polynomial coefficients **c**. These coefficients can be calculated by the evaluation of Eq. (8) with the known refraction index changes at all nodes of the thermal FE mesh. The Eq. (8) is then evaluated during the numerical solution of Eq. (6) and thus, the changes of the refraction index due to stress can be considered efficiently in the ray tracing. A more detailed description of the application of cylinder polynomials in the GRIN ray tracing is given in [19].

The result of the thermal FE analysis of the optical elements are the time-dependent nodal temperatures. The temperature changes *Δ**𝜗* lead to a spatial variation of the refraction index. The changes of the refraction index in a position *r*_{j} due to temperature changes can be described by

$$ \begin{aligned} \Delta n_{\mathrm{\vartheta},j} = &\frac{n_{{0}}-1}{2 n_{{0}}}\left(D_{0}\Delta\vartheta_{j}+D_{1}\vartheta_{j}^{2}+D_{2}\Delta\vartheta_{j}^{3} + \right. \\ &\left. \frac{E_{0}\Delta\vartheta_{j}+E_{1}\Delta\vartheta_{j}^{2}}{\lambda^{2}-\lambda_{\text{tk}}^{2}} \right)\;. \end{aligned} $$

(9)

Thereby, *D*_{0},*D*_{1},*D*_{2},*E*_{0},*E*_{1}, and *λ*_{tk} are material dependent constants and *n*_{0} is the refraction index at the reference temperature of 20^{∘}*C*. The spatially disturbed refraction index changes are then approximated with cylinder polynomials and evaluated during the ray tracing, such as previously described for the mechanical stresses. The refraction index changes due to stress and due to temperature changes are added during the ray tracing. Thus, mechanical stress effects and thermal effects are considerable in the ray tracing.

With the transformed results from the dynamical and the thermal system analysis the ray tracing can be performed. Thus, a holistic simulation of optical systems is possible with this method. Thereby, transient rigid body motions, dynamical deformations, refraction index changes due to mechanical stresses and thermally induced refraction index changes are considered computational efficiently during the ray tracing.

### Implementation

To investigate rigid body motion, deformation, mechanical stress and thermal effects in a dynamical-thermal-optical system, the introduced simulation procedure is implemented. The holistic simulation is performed using ANSYS for the building of the FE models, MatMorembs^{Footnote 1} for the MOR, Neweul-M^{2}^{Footnote 2} for the EMBS simulation, TheelaFEA for the thermal analysis, and OM-Sim^{Footnote 3} for the data transformation and the optical simulation. The latter four are MATLAB based software tools which are developed at the Institute of Engineering and Computational Mechanics at the University of Stuttgart. A holistic workflow diagram is shown in Fig. 1.

### Numerical example

The presented holistic dynamical-thermal-optical system simulation method is applied to a numerical example in the following. The used system and its excitation are shown in Fig. 2. It consists of a single lens with a diameter of 100 mm and a homogeneous refraction index of *n*_{0}=1.6 as long as no mechanical stresses and no temperature changes occur. The lens is fixed by a holding, which has six support beams evenly spaced around the circumference. The lens and the holding beams are modeled as elastic bodies, while the holding itself is considered to be rigid. As shown in the figure, the system is excitated mechanically with a predefined movement over the time in all spatial directions translational *a*_{t,x},*a*_{t,y},*a*_{t,z} and rotational *a*_{r,x},*a*_{r,y},*a*_{r,z}. Therefore, a time-dependent random signal based on a Gaussian distribution with a maximum translation of 1×10^{−3} mm, a maximum rotation of 1×10^{−2} rad, and a maximum frequency of 100 Hz is applied on the center of gravity of the optical system. A time-integration of the EMBS system leads to the deformations and rigid body motions of the lens and to the corresponding mechanical stresses. The thermal excitation are seven heat fluxes applied on the lens as shown in Fig. 2. They represent the heat transfer from seven light rays which come from infinity, propagate through the optical lens, and focus on the optical axis on the image plane if no disturbances exist. A time-integration for the thermal FE model leads to the temperatures over the time. For the analysis of the dynamical-thermal-optical system behavior, the results of the ray tracing algorithm are investigated next for the previously mentioned seven rays.

In order to investigate the dynamical-thermal-optical system for a large number of rays, another thermal boundary condition is applied. A heat flux is applied on all nodes of the thermal FE lens model which are within a cylinder with a radius of 20 mm, with the center at *x*=*y*=11 mm and with the height over the whole lens thickness. This applied thermal boundary condition is completely unrelated to the ray paths. It is assumed to visualize thermal influence on images only. The dynamical excitation is the same as described before. In this case the light propagates from one point uniformly disturbed through the lens and forms a grid on the image plane. So, a geometrical image simulation is performed. The unexcited system and its image are shown in Fig. 3.