## 3 Accelerated Structural Topology Optimization
### The Deep Learning-Assisted Topology Optimization (DLTOP) Methodology  
SIMP is one of the most widely employed, highly accurate and robust methodologies, applicable to a wide spectrum of TO problems. Nonetheless, in the field of STO, the required model scale, complexity and discretization level are continuously increasing, and thus the application of SIMP to such problems encounters bottlenecks, even in modern computing facilities, due to the substantial associated demand in computing time and resources. This insufficiency of SIMP motivated the development of DLTOP by the authors. In cases of pattern recognition problems, DBNs are able to discover the different levels of representation of the data nonlinearity. This capability of DBNs motivated the recent development of the so-called Deep Learning-assisted Topology Optimization (DLTOP) methodology by the authors of [1], which enables reducing the iterations required by SIMP. The novelty of DLTOP lies in the use of deep learning techniques (DBNs) to propose a close-to-final optimized configuration of a structural system, based upon the system configuration after only a few SIMP iterations. This eliminates the greatest portion of the required SIMP iterations to obtain the final optimized structural topology, thus resulting in a substantial reduction in the required computing time and resources, while overcoming potential bottlenecks that would otherwise be encountered. The novel DLMU methodology proposed in this paper partially relies on the idea of DLTOP; however, it is additionally assisted by information provided by reduced models of the design domain, thus allowing for further acceleration of SIMP. Before presenting the features and advantages of DLMU, it is useful to provide here a detailed description of DLTOP. DLTOP is a combination of SIMP and DBNs, where a DBN is trained to transform the pattern of density fluctuation of FEs generated during the starting iterations of SIMP to the final distribution of the density values \(x_{i}\) over the design domain. DBN prediction capability is built based on a training procedure performed once over benchmark TO problems. The main advantage of DLTOP is that DBN needs not be trained again before being applied to any STO problem, irrespective of the FE mesh configuration and type (structured/unstructured), domain dimensions, target density, loading and boundary conditions, SIMP implementation features (e.g., filter value), etc. This key feature of DLTOP is attributed to its implementation such that every FE is handled separately, without requiring information regarding its location over the domain, its specific boundary and loading conditions, etc. The validation of DLTOP via several benchmark topology optimization test examples indicates a reduction of the number of iterations originally required by SIMP by more than one order of magnitude. Expectedly, the gain in computing time offered by DLTOP is proportional to the TO problem size. DLTOP is described hereafter with the use of a qualitative example. Let us consider a design domain discretized with \(ne\) FEs. In every iteration \(t\) of SIMP, the density values \(x_{i}\) (that is also denoted as \(d_{i}\equiv x_{i}\)) are being updated for every FE. The fluctuation of the density value \(d_{i}\) of the \(i_{th}\) FE can be expressed as a function of \(t\) as follows:  
\[d_{i}=f(t)\qquad\forall\ i\in[1,ne] \tag{10}\]  
The density value fluctuation with respect to the SIMP iterations varies drastically for different FEs [1], due to their relative position over the domain with respect to loads, supports, etc. In this sense, each FE represents a different optimization history of density values with respect to the SIMP iterations, analogous to a discrete time-series. An initially uniform density value of 0.40 is specified for all FEs over the mesh, equal to the target volume ratio of the system of 40%; this constitutes common practice in SIMP implementation [26; 27]. The computational demand of SIMP depends on the number of FEs.