# Topology Optimization using LS-OPT®/Topology

**Topology Optimization using LS-OPT® /Topology**

**Author: Tushar Goel, LSTC, Livermore CA**

Topology optimization is a first-principle based optimization method to develop new concepts in engineering problems. Most previous studies in topology optimization have focused on designing linear structures with static loading conditions but there is relatively little work on handling non-linear problems involving dynamic loads, like those observed in crashworthiness optimization. The topology optimization in the context of impact analysis is a very complex problem due to non-linear interactions among material non-linearities, geometry, and the transient nature of the boundary conditions. Conventional methods are not practical for solving these non-linear topology optimization problems due to the high computational cost and the lack of sensitivity information.

A heuristic topology optimization method developed at the University of Notre Dame, known as hybrid cellular automata, has shown the most potential in handling topology optimization problems for crashworthiness problems. This method updates the design variable of an elements based on the objectives and constraint function information from its neighbors. No gradient information is required. The simplicity and effectiveness of this method for both two- and threedimensional problems has made it a suitable choice for implementation in LSOPT ® for topology optimization. This method has been applied to a host of linear and non-linear examples. The first application example is a statically loaded structure with linear behavior. While the bottom section is fixed, a uniformly distributed load is applied to the top. The geometry is meshed using 125k elements (1mm element size). Each linear-static analysis using the LS-DYNA implicit solver takes approximately 75s on a 2.66GHz Intel Xeon CPU with 4GB memory.

a) Initial geometry b) Final geometry c) IED d) von Mises-stress

Figure 1: Topology optimization of a statically loaded structure (125k elements).

The initial and final geometry obtained using the HCA method along with the resulting internal energy density (IED) and von-Mises stress contours are shown in Figure 1. The structure evolves to one with a homogenously material distribution and a reasonably uniform distribution of loads as depicted by the internal energy density and stress contours.

The initial geometry shown in Figure 1 is also analyzed by considering a dynamic load case when a pole hits the block with an initial velocity of 8.9m/s. The simulation is analyzed for 3ms. For this example, a refined mesh of one million elements (0.5mm element size) was used. The LS-DYNA® MPP explicit solver takes approximately 90 min on 32 processors of an SGI cluster to solve this problem. The final result, shown in Figure 2, clearly indicates the double-arch structure of the optimized shape and a relatively homogenous distribution of the internal energy density .

a) Final geometry b) Sectional view c) Internal energy density

Figure 2: Final geometry and internal energy density obtained using the HCA based topology optimization method for dynamically loaded structures (1 million elements).

While the results shown here provide a glimpse of the applicability of the topology optimization method available in LSOPT/ Topology, the capabilities of the tool also extend to handling

- non-box shaped domains,
- extruded design domains, and
- multiple loading conditions.

This topology optimization method is capable of handling large models incorporating both linear and non-linear structural analysis as applied to the industrial applications. The beta version of the LS-OPT®/Topology tool (without the graphical user interface) would be released by the end of April 2009. The first production version of the LSOPT ®/Topology tool including the

graphical user interface should be available by the end of December 2009.

#### More information on LS-OPT Topology:

www.lsoptsupport.com/documents/optimization/Goel_TopologyOpt_Salzburg2009.pdf/view