Integrated topology and packaging optimization for conceptual-level electric vehicle chassis design via the component-existence method

Abstract

Conventional vehicle architectures are undergoing significant transformation as automakers embrace electrification. With increased emphasis on lightweight structures design and efficient packaging of new electric powertrains, numerical tools are now essential to help solve these complex material and component distribution problems. To address these challenges, methods for integrated topology and packaging optimization (iTOPO) have been developed to couple these problem statements and form dynamic component-structure interactions. In this work, a component-existence approach is used for conducting iTOPO of self-contained electric vehicle chassis structures to demonstrate the benefits and scalability of this emerging methodology. Examples focus on incorporating simplified components for battery modules and electric motors within the underlying vehicle structure, integrating up to 43 components simultaneously in a 3D design domain. Here, discussion highlights the development of unique integrated layouts, methodology tunability, and practical insights of the formed component-structure interactions. iTOPO results are also compared to equivalent topology-only problems and show less than a 10% difference in compliance despite the addition of various complex integration requirements (e.g. multiple geometries, packaging symmetry).

Introduction

The automotive industry is experiencing a significant shift away from conventional internal combustion platforms and toward electric, autonomous, and adaptable vehicle architectures. Driven by policy demands for safer and more sustainable transportation, Canadian sales of zero-emissions vehicles (ZEVs) are expected to increase up to 10% of the current market by 2025 and up to 30% by 2030, with larger crossovers, sport utility vehicles (SUVs) and pick-up trucks showing strong growth potential in North America as indicated by the International Energy Agency (IEA). ¹ In response, original equipment manufacturers (OEMs) like Ford and General Motors (GM) are embracing electrification and seeking to increase their electric vehicle (EV) fleets by 40 new models in 2022 and 30 new models in 2025 respectively, as noted in the IEA’s 2021 report. ² To satisfy these targets OEMs are increasingly transitioning from conventional and dedicated vehicle platforms to more adaptable skateboard configurations that integrate energy storage and powertrain systems into a common chassis structure. Examples include Ford’s electric body-on-frame concept as well as the GM’s Ultium platform, enabling tunability of vehicle body and interiors without significant change to the core structure. As automakers continue to develop these modular and flexible platforms design for cost-effective vehicle range is of particular importance, and poses unique challenges related to the structural mass and packaging efficiency of components into the system-level architecture. That is, to reduce EV costs and extend the range of traditional skateboard platforms new designs must seek to maximize efficiency of material distribution and utilization, ultimately incorporating components such as battery cells and motors into primary structural members. While these concepts have been discussed in literature (e.g. see Zhang et al. ³ ) they have been increasingly popularized for commercial EVs. At the conceptual-level, development of these new platforms can be broken down into material and component distribution problems and offers the ideal application of numerical methods for structural topology optimization (TO) and component packaging optimization (PO). An overview of these methods is provided next, followed by a review of integrated topology and packaging optimization (iTOPO) to support more holistic designs featuring efficient component-structure interactions. Here, additional emphasis is provided with respect to methodology roles and applications within the automotive industry.

Topology optimization

Topology optimization represents a class of numerical techniques for solving material distribution problems typically associated with lightweight structural design. In most commercial software and practical applications TO is enabled using finite element discretization and analysis coupled with specialized material interpolation techniques and general non-linear programing. Broadly speaking, this describes the density-method for topology optimization formalized in the monograph by Bendsøe and Sigmund ⁴ and applied within the current work. Key advancements for commercial TO programs today include the refinement of interpolation schemes, filtering schemes, as well as introduction of increasingly robust gradient-based optimizers like the method of moving asymptotes.^5,6 Additional TO implementations and developments are discussed in the review papers by Rozvany ⁷ and Sigmund and Maute. ⁸

Following its maturation and adoption in commercial programs topology optimization has seen various applications within the automotive industry. At the part and assembly levels, TO has been applied within conventional architectures to various sub-assemblies such as control arms, ⁹ engine cradles, ¹⁰ cross members, ¹¹ and door structures ¹² to name a few. In these applications, TO is typically used to develop alternative designs for maximizing stiffness (minimizing compliance) under a given lightweighting requirement, or to minimize mass while maintaining the performance from a baseline structure (displacement or frequency-based constraints). This also includes expanded methods focused on design for manufacturability. System-level case studies have also been conducted, with early automotive implementations focused on full vehicle chassis and body structures as in Yang and Chahande ¹³ and Wang et al. ¹⁴ More recent applications include the development of high performance space frames in Cavazzuti et al. ¹⁵ as well as extensions to multi-material topology optimization to further improve lightweighting potential in automotive applications by Li and Kim. ¹⁶

While these studies demonstrate practical implementations of TO as well as various extensions to standard methodology for improving design development (e.g. manufacturing constraints, multi-material, etc.) consideration of component-structure interactions is largely omitted from existing workflows. Within these problems TO is often restricted to pre-defined design domains that deliberately ignore component integrations from structural consideration and suggests an opportunity for iTOPO methods to develop more holistic design insights. This is especially important with respect to modern EVs as designers seek to efficiently incorporate key components like battery cells (/module housings) within the underlying vehicle architecture. ³

Packaging optimization

Packaging optimization represents a class of numerical design tool for solving component distribution problems and generally consists of three noted areas of focus:component-to-domain, component-to-component, and component-to-system. While generally limited to research-level programs and applications PO problems have been addressed using a variety of implementations. These are most commonly heuristic methods as in He et al. ¹⁷ or standard non-deterministic methods like genetic algorithm (GA) as in Miao et al. ¹⁸ along with analogy-based concepts like the rubber band method by Dong et al. ¹⁹ and the vector fields method by Carrick and Kim. ²⁰ Generally, these techniques are tailored for addressing two out of the three noted packaging scenarios, with differences in the treatment of objective and constraint functions and the inclusion of system responses like center of gravity (e.g. Dandurand et al. ²¹ ).

Select vehicle case studies have been examined using PO methods. For example, Miao et al. ¹⁸ explored a system-level vehicle configuration problem considering the positioning of driveline components in a mid-size truck using GA. Another example by Dandurand et al. ²¹ explored the packaging of components in a vehicle engine bay. Examples of packaging optimization considering system performance have also been considered, with work largely restricted to thermal management and analysis (see Pitre and Kim ²² in the case of a rover platform).

While these examples demonstrate applications of PO in vehicle design, they involve packaging components around pre-defined structures and domains that prioritize geometric considerations only (e.g. overlap avoidance, center of gravity). That is, standard PO techniques tend to ignore structural integrations from the packaging design process and thus exclude potential lightweighting from the problem statement altogether. As with standard TO, this suggests an opportunity for iTOPO methods to develop more holistic design insights and component-structure interactions in EV-based design problems where lightweighting has a critical impact on range and performance.

Integrated topology and packaging optimization

Recognizing the potential benefit of coupled TO and PO, iTOPO represents a class of techniques that solves the material and component distribution problems simultaneously. These methods seek to dynamically consider structural TO and PO, allowing for the development of component-structure interactions during the design process and increasing the effective design freedom from each respective sub-problem. Conventional iTOPO methods are based on the standard TO workflow and coordinates material and component distribution variables (material pseudo-densities $\underset{~}{\hat{x}}$ and component position variables $\underset{~}{\hat{\varphi }}$ ). These can be further categorized based on their specific treatment of components and their form of integration encompassing four main approaches (interpolation, direct, level-set, and multi-point constraints), whose application within academic and industry case studies are reviewed further below.

Conventional interpolation methods were developed first by Qian and Ananthasuresh ²³ and revisited by Xia et al. ²⁴ where components are modeled via manipulation of local material properties and coupled with the base topology through the existing interpolation infrastructure. Despite their early adoption these approaches are primarily restricted to 2D academic examples, with some practical extensions in 3D applied to the design of bridge structures with pre-defined moveable supports. Direct methods were developed next, introduced by Zhu et al. ²⁵ and with early extensions by Zhu et al. ²⁶ Here, components are treated as meshed bodies and coupled with the base topology through node-to-node transition elements. Examples of direct approaches are also generally restricted to 2D problems given the requirement for iterative remeshing procedures, however some examples include practical applications for aerospace components as in Zhu et al. ²⁵ and Qiao et al. ²⁷ Level-set methods were developed by Zhang et al. ²⁸ with additional work by Xia et al. ²⁹ Like the general interpolation method components here are effectively modeled at the material property level, but now using implicit level-set functions. Despite offering alternative forms of overlap-avoidance and numerical efficiency, level-set methods are only demonstrated in 2D and 3D academic models. Today, the dominant methods and applications for iTOPO utilize the multi-point constraint approach (MPC). Here components are treated as explicitly meshed entities but are now coupled to the base topology domain through displacement constrained relationships (MPC equations), effectively establishing rigid joints between discrete locations on the component and the underlying mesh. These methods were introduced in Zhu et al. ³⁰ with various practical extensions developed for aerospace applications. This includes component-structure optimization within a fuselage bulkhead, ³¹ and integration of batteries within an unmanned aerial vehicle. ³²

In addition to these dominant treatments are alternative iTOPO strategies that maintain the use of coupled pseudo-densities $\underset{~}{\hat{x}}$ and position variables $\underset{~}{\hat{\varphi }}$ while incorporating new solutions processes. This includes techniques such as the morphable mesh method by Bakhtiarinejad et al. ³³ or method of moving morphable components by Wang et al. ³⁴ Beyond these treatments are other emerging methods within the field that replace the use of explicit position-based variables with coupled material and component pseudo-densities ( $\underset{~}{\hat{x}}$ and now $\underset{~}{\hat{Z}}$ ). Here, a series of component candidates $(k=1,...,n_c)$ are defined within the base topology at various locations and orientations prior to optimization. With this framework, continuous 0-1 component pseudo-densities ${\hat{z}}_k$ are used to describe a state of existence for each candidate and ultimately specify which should remain within the model according to an integration constraint $\overline{Z}$ . These are referred to as component-existence models and were first presented by Roper and Kim ³⁵ considering 2D academic models and an automotive bumper structure, and later extended into 3D. ³⁶

Knowledge gap and objectives

While methods for iTOPO have matured, their application within system-level automotive design has largely been neglected despite their clear utility toward industry trends like electrification and the need for more integrated powertrain and chassis structures. Instead, the development of many new EV architectures continues to rely on more manual and standalone applications of TO and PO methods that retain more conventional ladder-frame structures and ignore the potential for efficient component-structure interactions. Correspondingly, the objective of this paper is to explore the more holistic development of flexible self-contained EV platforms considering structural topology and powertrain components simultaneously via a robust optimization framework. This goes beyond existing part- and assembly-level optimization studies to more effectively address the integration problem between EV components, such as battery and motor models, with the underlying vehicle structure using a broader system-level approach. This is explored in two conceptual design problems that include various complex integration requirements, such as packaging constraints like symmetry and packaging of multiple unique geometries. This is achieved using an advanced component-existence model to couple the typically standalone TO and PO problem aspects and methodology capabilities into a single-loop solution process that effectively and efficiently produces design options and insights. This work goes beyond various position-based methods and utilizes concepts from Roper and Kim^35,36 for coordinating material and component pseudo-densities in a unique and inherently scalable framework capable of exploring the design domain and generating various concepts for further investigation.

The remainder of this paper begins with an overview of the component-existence model. This is followed by numerical examples, discussion and overall summary of results.

Theory overview

Problem statement

Component-existence models are primarily based on standard TO workflows, and rely on finite element discretization for system modeling and general non-linear optimizers for conducting design updates. A typical problem statement is shown in equation (1), which contains dependencies on explicit material and component design variables ( $\underset{~}{\hat{x}},\underset{~}{\hat{Z}}$ ) as well as element-wise representations referred to as material and component design fields ( $\underset{~}{\hat{x}},\underset{~}{\hat{Z}}$ ). Transformation from variables to fields is achieved via forward mapping functions $\underset{~}{x}={\Phi }_1(\underset{~}{\widehat{x}})$ and $\underset{~}{z}={\Phi }_2(\widehat{\underset{~}{z}})$ which convert the dimensionality of problem variables to a standard element-wise representation in ${\mathbb{R}}^{n_e}$ for subsequent coupling to form the component-structure interaction. These coupled variables are then used for property interpolation within a modified 0-1 SIMP model and ultimately assembled within a FEA function call solving $\underset{~}{K}(\underset{~}{x},\underset{~}{z})\underset{~}{u}=\underset{~}{f}$ . As a result, analysis and select problem statement responses are based on material and component design fields that describe the underlying structural topology (solid and void elements $e,e=1,...,n_e$ ) and component layout (solid and void elements e describing component candidates $k,k=1,...,n_c$ ). While $g_0$ and $g_i^{(1)}$ feature design field dependencies to describe typical responses such as compliance and mass fraction, the second constraint class $g_j^{(2)}$ is reserved for describing responses with explicit relationships to $\underset{~}{\hat{x}}$ and $\widehat{\underset{~}{z}}$ which can be used to represent packaging aspects of the problem such as integration status of component candidates ${\hat{z}}_k$ .

Beyond these fundamental problem aspects are additional design fields and backward mapping functions to complete the numerical framework. This includes fields for describing the component candidate locations and methods for converting element-wise field sensitivities $\partial (•)/\partial x_e$ and $\partial (•)/\partial z_e$ to variable sensitivities $\partial (•)/\partial {\hat{x}}_e$ and $\partial (•)/\partial {\hat{z}}_k$ to drive the design updates. A complete review of these techniques is provided in Roper and Kim,^35,36 with an overview of the solution process provided next.

$$\begin{array}{l}\mathrm{minimize}(\widehat{\underset{~}{x}},\widehat{\underset{~}{z}}):g_0(\underset{~}{x},\underset{~}{z},\underset{~}{u})\\ \mathrm{subject}\mathrm{to}:g_i^{(1)}(\underset{~}{x},\underset{~}{z},\underset{~}{u})\le 0\\ :g_j^{(2)}(\widehat{\underset{~}{x}},\widehat{\underset{~}{z}})\le 0\\ \mathrm{where:}\underset{~}{K}\underset{~}{u}=\underset{~}{f}\\ :x_e={\Phi }_1(\underset{~}{\widehat{x}}),z_e={\Phi }_2(\widehat{\underset{~}{z}})\\ :\forall \mathrm{element}e,{\widehat{x}}_e\in [0,1]\\ :\forall \mathrm{candidate}k,{\widehat{z}}_k\in [0,1]\end{array}\}$$ (1)

Solution process

A 2D example is provided in Figure 1 to demonstrate the component-existence model’s solution process. In its simplest form, this begins with a standard problem definition with respect to the base topology domain ${\mathrm{\Omega }}_b$ , as well as the initialization of component candidate domains ${\mathrm{\Omega }}_c^{(k)}$ (in this case, $n_c=32$ non-overlapping square components across the design domain). During optimization, the component-existence model drives development of the material existence field ( $x_e\in [0,1]$ , shown using a gray color scale) and component existence field ( $z_e\in [0,1]$ , shown using a blue color scale), that combine to form the coupled field ( ${\eta }_e\in [0,2]$ , shown in gray-blue color scale). While Figure 1 demonstrates a typical design progression including intermediates, elements ultimately belonging to a discretized component candidate are shown in blue while elements belonging to the base topology are shown in black. Here a solid element may be defined by the base topology only (black), a component (blue), or some combination of the two (also shown in blue). Numerically, this coupling is achieved via the model in Roper and Kim, ³⁵ which is then truncated within 0-1 bounds ${\tilde{\eta }}_e$ for compatibility with a standard SIMP interpolation framework as in equation (2). This ultimately describes the component-structure interactions and where 0-1 represent “void” and “solid” properties. This is then used for analysis to obtain key system responses (e.g. compliance and mass fraction).

$$E_e=({\tilde{\eta }}_e{)}^p(E^{(1)}-E^{(2)})+E^{(2)}$$ (2)

Figure 1.

Sample iTOPO solution process using a component-existence model. This shows the definition of square components initialized in $n_c=32$ candidate locations across the domain without overlaps and the progressive generation of material and component existence fields to ultimately develop efficient component-structure interactions.

From this treatment and considering the general problem statement in equation (1), component-existence models share parallels with traditional manufacturing constraints in TO (e.g. Vatanabe et al. ³⁷ ) that redistribute material within the domain according to a given design guideline (e.g. constant cross section to model an extrusion process). In this case however, the material redistribution is executed to satisfy a component integration requirement (i.e. number of components to keep) and effectively reserves part of the design domain to form the iTOPO component geometries in discrete locations and orientations. Beyond this solution process are various other practical extensions to iTOPO possible within component-existence frameworks. This includes the development of multi-field candidate initializations (i.e. component initializations where candidates may partially overlap one another) to increase the packaging design freedom and multi-phase components to expand modeling features (i.e. components whose structure features both solid and void material phases, representing features like hollow interiors and material keep-out-zones). This also extends to multi-component problems (i.e. problems featuring more than one unique geometry in the same design domain), as well as symmetry packaging constraints and regional packaging constraints. Ultimately, combining these various problem aspects represents the culmination of iTOPO capabilities for broader and more meaningful system-level design studies and is the focus of the industry applications presented in this work.

Numerical examples

To develop more holistic self-contained electric vehicle platforms beyond use of standalone TO and PO methods, the component-existence method is applied in two vehicle studies to obtain various conceptual-level designs. A 2006 Ford F250 is used as the reference platform, with the present work examining the replacement of a conventional ladder frame configuration shown in Figure 2(a) with an EV skateboard chassis structure with embedded components shown in Figure 2(b). These are referred to as Chassis (1) and Chassis (2), with the former exploring iTOPO of battery housings within the structural design domain, and the latter extending this study to consider component models of front and rear drive motors. Results are provided for iTOPO as well as free topology cases (fTOs or simply TO, problems solved using traditional topology optimization methods) to show designs obtained from equivalent studies without integration requirements $({\overline{Z}}_l=0)$ . Designs are also discussed with respect to various insights afforded by the component-existence model, including alternative solutions via problem parameter tuning and design interpretation.

Figure 2.

Case study overview: (a) reference configuration and (b) iTOPO design space configuration. Replacement of a conventional ladder frame chassis for integration of electric vehicle powertrain components (battery housings and motors).

Chassis (1)

This case study explores the lightweight design of an EV chassis with integrated modular battery housings. Incorporating these components as structural members within the chassis offers a more efficient treatment against simple addition to a conventional ladder frame configuration and enables immediate benefits such as parts consolidation and design simplification. This treatment also has important implications toward decreasing the effective battery costs which are now shared with the broader vehicle structure. Based on the current methodology, battery housings are assumed to share the same material properties as the base topology and follows the earlier analogy of iTOPO as a manufacturing constraint in TO.

Referring to the iTOPO setup, this case study is examined under both single-field component initializations (i.e. where candidates do not intersect one another) and multi-field component initializations (i.e. where candidates intersect one another in the initial setup), shown in Figure 3(a) and (b) respectively. Here, battery housings are modeled as slender rectangular members with a structural exterior and interior voids (where the actual non-structural cells may be incorporated), describing a multi-phase component model (i.e. use of both solid and void material phases to describe the component). In the single-field study these are placed in $n_c=48$ length-wise candidate locations shown in Figure 3(a), while width-wise overlapping candidates are included in the multi-field case with $n_c=96$ shown in Figure 3(b). In both cases, the battery housings are initialized toward to center of the overall design domain (under the passenger cab). The remainder of the problem setup is given in Figure A1, showing the broader 3D design domain with ∼44,000 designable elements, ∼500 non-designable elements (body mounts, suspension mounts), and $n_s=2$ subcases imparting global chassis twist.

Figure 3.

Chassis (1) problem definition: (a) battery housings (solid-void component candidates) with $n_c=48$ and no overlaps and (b) battery housings (solid-void component candidates) with $n_c=96$ and overlaps.

The problem statement for this case study is shown in equation (3), which seeks to minimize weighted compliance considering a mass fraction constraint of $\overline{M}=0.50$ and integration requirement of ${\overline{Z}}_1=28$ (with symmetry). This enforces the number of component candidates to keep in the final optimum solution.

$$\begin{array}{l}\mathrm{minimize}(\widehat{\underset{~}{x}},\widehat{\underset{~}{z}}):\sum _{i=1}^{n_s}{w}_i{C}_i\\ \mathrm{subject}\mathrm{to}:M\le \overline{M}\\ :{\overline{Z}}_1\le Z_1\left(\mathrm{with}\mathrm{symmetry}\right)\\ \mathrm{where}:\underset{~}{K}\underset{~}{u}=\underset{~}{f}\\ :x_e={\Phi }_1(\underset{~}{\widehat{x}}),z_e={\Phi }_2(\widehat{\underset{~}{z}})\\ :\forall \mathrm{element}e,{\widehat{x}}_e\in [0,1]\\ :\forall \mathrm{candidate}k,{\widehat{z}}_k\in [0,1]\end{array}\}$$ (3)

Referring to the general problem statement in equation (1), the objective function is given as $g_0=\sum _i^{n_s}{w}_i{C}_i$ , where $w_i$ is a weighting factor for subcase i $(w_i=1)$ and $C_i$ is the total compliance for subcase i $(C_i={\underset{~}{u}}^T\underset{~}{K}\underset{~}{u})$ . Here, the first constraint class $g^{(1)}$ describes the lightweighting criteria in the problem, where $M$ represents the current mass fraction of the designable elements shown in equation (4) and $\overline{M}$ describes the upper bound of the constraint. The second constraint class $g^{(2)}$ describes the packaging aspects of the problem, with $Z_l$ describing the active number of candidates present in the domain shown in equation (5) and ${\overline{Z}}_l$ representing the user-defined integration requirement (i.e. the number of components to keep) for geometry $l(l=1,...,n_l).$

$$M=\frac{\sum _{e=1}^{n_e}{M}_e}{\sum _{e=1}^{n_e}{V}_e{M}_e^{(1)}}$$ (4)

$$Z_l=\sum _{k=1}^{n_c^{(l)}}{\hat{z}}_k^{(l)}$$ (5)

In equation (4), $M_e$ is the interpolated element density as a function of $x_e$ and $z_e$ field quantities based on the coupling and 0-1 SIMP framework in Roper and Kim. ³⁵ $V_e$ represents the volume for element e, and $M_e^{(1)}$ represents the density of element e with solid nominal properties. In equation (5), ${\hat{z}}_k^{(l)}$ represents the component pseudo-density for candidate $k(k=1,...,n_c^{(l)})$ associated with geometry $l$ .

Solutions to this problem are provided in Figure 4(a) to (f) showing single-field, and multi-field iTOPO results respectively. Considering the single-field solution only, Figure 4(a) provides a visualization of the final component layout, (b) a visualization of the component structure interaction, and (c) a section view of the component-structure interaction to visualize internal features. As previously noted, solid elements belonging to structural component candidates are shown in blue while solid elements belonging to the base topology are shown in black. Void elements are filtered out of the result visualization, and elements shown in red are simply used to visualize intersection with the cut plane. This display scheme is repeated throughout the paper.

Figure 4.

Chassis (1) numerical results: (a–c) single-field study with non-overlapping multi-phase component candidates and (d–f) multi-field study with overlapping multi-phase component candidates.

Considering performance, the chassis compliance for the iTOPO cases naturally increase compared to the fTO solution in Figure A3(a) to (c) due to the reduced model design freedom (i.e. integration requirements). However, its noted that these increases are limited within reasonable bounds, to only 5.4% in the single-field model and only 4.8% in the multi-field model. Next, considering the overall material and component distributions the iTOPO designs feature the generally expected closed hollow box structure, effectively maximizing the chassis polar moment of inertia against the global twisting loadcase and closely matching the fTO solution. Internal reinforcement features are also noted along the vehicle centerline at the middle of the cab, as well as added material placement toward the front of the chassis. Comparing the single- and multi-field implementations, components are aligned along the external edges of the domain with a select distribution of internal components to effectively form a large chassis cross member. In the single-field case, 16 components are used to form external rails while 12 components are used to form the internal reinforcement. These regions are subsequently arranged in a checkerboard-like pattern that maximizes the component distribution while their connections are supported by additional material from the base topology. This is shown in more detail in Figure 4(c). In the multi-field case, only 10 components are needed to form an internal support structure that connects directly with the remaining components again forming rails. This leverages the use of width-wise components to more readily span the design domain and refine the size of the internal structures.

Overall, the results shown here demonstrate the development of intuitive and holistic component-structure interactions from non-intuitive initial design domains. Here, application of iTOPO enables effective exploration of the design space and considers various practical aspects such as use of single- and multi-field initializations, multi-phase (solid-void) components, and packaging symmetry without a significant increase in computing and design cycle times compared to fTO studies. Ultimately, considering structural and packaging aspects from a system-level perspective results in unique insights and options ideal for conceptual design problems featuring complex integration requirements.

Design insights

Beyond the primary results in Figure 4 are various secondary solutions attainable through methodology tuning parameters. One mechanism from Roper and Kim ³⁵ is referred to as $\alpha$ tuning, which adjusts the backward mapping function when generating surrogate sensitivities for $\partial (•)/\partial {\hat{z}}_k$ from $\partial (•)/\partial z_e$ with respect to the objective and class one constraint functions. Inspired from sensitivity filtering techniques, this mapping function is shown in (6) and provides a method for exploring the domain to produce various local optimum solutions as in Figures 5 and 6 $(0<\alpha <1).$

$$\frac{\partial (•)}{\partial {\hat{z}}_k}=\frac{1}{\sum _{e\in {\mathrm{\Omega }}_c^{(k)}}|\alpha -x_e|}\sum _{e\in {\mathrm{\Omega }}_c^{(k)}}|\alpha -x_e|\frac{\partial (•)}{\partial z_e}$$ (6)

Figure 5.

Secondary iTOPO solutions for the single-field Chassis (1) problem using $\alpha$ tuning (backward mapping tuning variable): (a–c) $\alpha =0.1$ , (d–f) $\alpha =0.5$ , and (g–i) $\alpha =0.9$ .

Figure 6.

Secondary iTOPO solutions for the multi-field Chassis (1) problem using $\alpha$ tuning (backward mapping tuning variable): (a–c) $\alpha =0.1$ , (d–f) $\alpha =0.5$ , and (g–i) $\alpha =0.9$ .

Here, the resulting secondary solutions continue to show parallels with the primary loadpaths in the fTO solution as well as common integration features between each iTOPO design (e.g. chassis rails or cross members). Outside of these core similarities each design features a distinct bulk distribution of component candidates in the model, from tightly packed configurations in Figures 5(a) to (c) and 6(a) to (c) to the development of more conventional cross member structures in Figures 5(g) to (i) and 6(g) to (i). These readily available and unique design alternatives are particularly useful during conceptual-level trade-off studies, as new requirements are added to the problem beyond stiffness and lightweighting criteria.

Chassis (2)

As an extension to Chassis (1), the initial problem from Figure 3 is expanded to consider a secondary battery pack location along with integration of front and rear drive motors as shown in Figure 7. As before, incorporating multiple key driveline components within the broader chassis domain offers a holistic system-level design strategy that maintains the previously noted benefits such as parts consolidation and effective cost reduction. In addition to these benefits are further designer-focused controls and insights afforded by the expanded iTOPO domain. For example, this study applies an effective regional packaging constraint to the added battery housings included in the model, controlling their candidate locations in the model as well as the integration requirements for these components. Here, despite their equivalent geometric and functional representation to members in the primary pack, these added housings are treated as a distinct new set of component candidates with their own integration requirements, enabling designers to adjust and study $\overline{Z}$ constraints for developing vehicle platform variants (e.g. “extended-range option”). Another noted practical advantage is design interpretation flexibility during iTOPO result post-processing. For example, as electric motors are integrated within the overall chassis structure any resulting component-structure interactions can be projected either into the full chassis design or back into the detailed motor housing/mount design itself. With this flexibility, designers can drive informed changes to entire systems as well as individual components.

Figure 7.

Chassis (2) problem definition. Primary battery housings with $n_c^{(1)}=96$ , secondary battery housings with $n_c^{(2)}=36$ , rear drive motors with $n_c^{(3)}=14$ , and front drive motors with $n_c^{(4)}=4$ . Battery housings contain internal voids(i.e. hollow) and drive motors contain external and internal voids (i.e. hollow and material keep-out-zones).

Considering the iTOPO problem definition, this case study is examined under multi-field multi-component initializations with solid-void part models. Referring to Figure 7, the structural battery housings are again modeled as hollow rectangular members and initialized in the distinct primary and secondary regions as mentioned above. The first is the same multi-field definition from before with $n_c^{(1)}=96$ candidates in overlapping length- and width-wise orientations. The second is a separate multi-field definition extending from the primary pack toward the rear of the truck frame. Here, battery housings are initialized in both orientations as well as stacked to make up $n_c^{(2)}=36$ candidates. Rear drive unit pairs are incorporated along the length of the truck bed toward the rear wheels and are simplified to represent motor housings with internal voids (i.e. hollow cylindrical members). This component definition also includes external voids to ensure access to the front and rear faces (e.g. output shaft access) and are initialized in $n_c^{(3)}=14$ overlapping locations. Lastly, a larger forward drive unit is modeled at the front of the vehicle adopting the same housing analogy as the rear motors. This also consists of both internal and external voids and is initialized in $n_c^{(4)}=4$ overlapping candidate locations. Additional design domain details are provided in Figure A2. This features the same mesh discretization from Chassis (1), but with additional body-mount bending loads applied as a third subcase. This results in $n_s=3$ subcases imparting global chassis twist and global chassis bending.

The problem statement for this study is shown in equation (7). Like Chassis (1), this features a weighted compliance objective function and mass fraction constraint of $\overline{M}=0.50$ . Unlike the Chassis (1) study outlined in Figure 3, this features multiple unique geometries and various integration requirements, where ${\overline{Z}}_1=28$ (with symmetry), ${\overline{Z}}_2=12$ (with symmetry), ${\overline{Z}}_3=2$ (with symmetry), and ${\overline{Z}}_4=1$ . As before, $w_i=1$ for all subcases.

$$\begin{array}{l}\mathrm{minimize}(\widehat{\underset{~}{x}},\widehat{\underset{~}{z}}):\sum _{i=1}^{n_s}{w}_i{C}_i\\ \mathrm{subject}\mathrm{to}:M\le \overline{M}\\ :{\overline{Z}}_1\le Z_1\left(\mathrm{with}\mathrm{symmetry}\right)\\ \mathrm{where}:\underset{~}{K}\underset{~}{u}=\underset{~}{f}\\ :x_e={\Phi }_1(\underset{~}{\widehat{x}}),z_e={\Phi }_2(\widehat{\underset{~}{z}})\\ :\forall \mathrm{element}e,{\widehat{x}}_e\in [0,1]\\ :\forall \mathrm{candidate}k,{\widehat{z}}_k\in [0,1]\end{array}\}$$ (7)

Solutions to this problem are shown in Figure 8, with the fTO also given in Figure A3(d) to (f) in the Appendix for comparison and reference. Like Chassis (1), the compliance of the iTOPO design increases compared to the fTO case (9.7%); however, its noted that this increase is again restricted within reasonable bounds even despite the inclusion of additional integration requirements in the design domain. Similarly, the overall material and component distributions in Figure 8 feature the expected closed hollow box structure due to the global chassis torsion, however with additional internal members and general reinforcement now biased rearward following the addition of body-mount bending loads. This is also reflected in the fTO solution. Overall, components are distributed with further parallels to the fTO solution and earlier case studies, featuring predominantly length-wise battery housings making up the core chassis rails supported by the width-wise candidates to the rear of the cab forming cross members. Beyond the core structure surrounding the primary and secondary battery housings are the integrated drive units. Here, the rear pair are integrated in the forward-most candidate positions to couple with the material distribution in the area while the forward drive unit integrates in the rearward most position under similar conditions. The resulting motor mounts and chassis structures show strong parallels to the original fTO material distribution.

Figure 8.

Chassis (2) numerical results: (a–d) multi-field, multi-component study with solid-void component candidates.

Overall, this case study represents the culmination of various iTOPO requirements, with the associated results demonstrating potential uses, insights, and problem configurations available from component-existence methods. Building from Chassis (1), the problem statement and setup readily adapts to include new geometries and domain controls. This includes multi-component aspects such as regional packaging constraints with respect to physical integration sites in the model as well as decoupling of like-geometries in the problem statement (i.e. secondary housings with separate $\overline{Z}$ requirements). This also extends to new component features with the application of external void elements as part of the multi-phase motor housing definitions. This goes beyond simple treatment of hollow members to define external material keep-out zones during integration, resulting is additional material redistributions and design insights discussed next. Like before, this example demonstrates the development of a unique vehicle configuration featuring intuitive structural and component packaging layouts obtained from non-intuitive multi-component packaging requirements. The resulting performance is comparable to baseline fTO structures, and suggests iTOPO as an effective tool for use in conceptual-level design studies.

Design insights

Beyond the macro-level iTOPO outcomes from this study, closer examination around the motor housing integrations can demonstrate the extended insights and modeling advantages available via the component-existence model. Here, two features are of note from Figure 9. The first is the component definition and application of external voids on the left and right faces of the motor housing. Representing keep-out-zones, this definition extends the component description beyond just its structural configuration to model practical accessibility requirements (e.g. access to the motor output shaft). This relates to the second noted feature, with broader implications on available iTOPO design insights. Here, component structure interactions are developed around the external-void definition to establish discrete motor mounts on the main housing body that ultimately seek to increase overall global chassis stiffness. In this sense, the iTOPO model describes a method for developing efficient joints between components and the base structure. Other interpretations of the resulting component-structure interaction include the design of chassis features according to the prescribed layout, or alternatively as motor housing features by projecting these insights back onto the component level.

Figure 9.

Chassis (2) review of a component-structure interaction showing progressive addition of connecting elements form the base topology to the front motor housing: (a) component only and (b–d) progressive layers of connecting adjacent elements.

In each case, application of iTOPO enables flexible generation and treatment of unique results aligned with the previously noted concepts of structurally efficient integration and parts consolidation for reduced costs and complexity. This goes beyond independent TO and PO treatments, with benefits also realized without significant computing costs compared to the fTO solution, offering a practical methodology for use in automotive design.

Summary and conclusion

This work explores the conceptual-level design of self-contained EV platforms using an iTOPO framework for developing holistic component-structure interactions between structural topology and powertrain components. With an emphasis on system-level architectures and EV design, these problems couple multiple integration functions simultaneously, first incorporating aspects like multi-phase components in multiple candidate orientations. This then expands to include more advanced problems with up to three unique geometries, packaging constraints like integration symmetry, and extended component features like the application of external-void elements to model material keep-out zones. Overall, each problem discusses the unique results, designer insights, and model interpretations, while demonstrating practical aspects of the underlying iTOPO methodology. In Chassis (1) battery housings are incorporated into the structural domain of a vehicle model, providing a parts consolidation solution compared to simple addition within a ladder frame chassis. Here, methodology tunability is also discussed and used to develop various alternative chassis concepts. This case study is expanded in Chassis (2) with the addition of secondary battery housings and drive units, providing further system- and part-level insights on potential vehicle variants (e.g. control over secondary battery cells) or component modifications (e.g. adjusted motor mounts for seamless chassis integration). In both cases designs are compared to the equivalent fTO solutions, with the component-existence model restricting performance losses within 10% despite demanding integration requirements and restrictions on overall material design freedoms.

Beyond development of unique conceptual platforms, the examples developed here demonstrate the continued advantages of the component-existence model and its ability to develop efficient component structure interactions even as problem complexity scales. In addition to the core framework applied here are opportunities to further enhance the practicality of component-existence models. Here, future work may include the expansion of system responses in the optimization problem statement beyond compliance minimization and mass fraction constraints. Of particular interest for practical design are vibration-based responses via normal modes analysis (e.g. first natural frequency), as well as thermal analysis (e.g. thermal compliance) and material self-weight. These extensions may be studied with minimal modification to the component-existence model, and continue to rely on the existing framework for coupling, material interpolation functions, and mapping functions. While this offers a path forward for more complex responses, additional heuristics may be required to overcome any numerical issues, as well as general model updates to extend capabilities. For example, the coupling and interpolation scheme used in equation (2) may be updated to include multi-material modeling such that components and the base topology may adopt independent properties (e.g. aluminum components in a steel base TO domain). Ultimately, with these extensions the component-existence model can act as a foundation for addressing increasingly complex iTOPO problems and develop efficient component-structure interactions driving lightweight design.