A system-level computational model of a recently patented and prototyped novel steam engine technology was developed from first principles for the express purpose of performing design optimization studies for the engine's inventors. The developed system model consists of numerous submodels including a flow model of the intake process, a dynamic model of the intake valve response, a pressure model of the engine cylinder, a kinematic model of the engine piston, and an output model that determines engine performance parameters. A crank-angle discretization strategy was employed to capture the performance of engine throughout a full cycle of operation, thus requiring all engine design submodels to be evaluated at each crank angle of interest. To produce a system model with sufficient computational speed to be useful within optimization algorithms, which must exercise the system level model repeatedly, various simplifying assumptions and modeling approximations were utilized. The model was tested by performing a series of multi-objective design optimization case studies using the geometry and operating conditions of the prototype engine as a baseline. The results produced were determined to properly capture the fundamental behavior of the engine as observed in the operation of the prototype and demonstrated that the design of engine technology could be improved over the baseline using the developed computational model. Furthermore, the results of this study demonstrate the applicability of using a multi-objective optimization-driven approach to conduct conceptual design efforts for various engine system technologies.

## Introduction

The design and development of engine technologies has been the subject of countless papers and publications over the years. This is because the generation of useful energy is essential to human society, and technologies that can generate useful energy as efficiently as possible are extremely compelling given the benefits of reducing waste while simultaneously increasing available power. Most power conversion technologies, such as the internal combustion engine, are quite mature and current research efforts in these areas are focused on making marginal performance and efficiency gains through better understanding of detailed aspects of the design and operation of the engine technologies considered, and many of these improvements are realized using high fidelity modeling and design optimization techniques [1–4]. Modeling and design optimization are numerical techniques used to systematically evaluate a large number of potential combinations of system design parameters in a computational environment in and order to ensure the best system performance possible.

Optimization techniques can be used on practically any design problem, provided computational models for a system of interest exit. For example, recent studies that use modeling and optimization in the area of energy system design include a study focused on the design and analysis of wave energy conversion systems [5], and another research effort involving the modeling and design of large scale energy supply systems [6]. Due to the continued interest in the economical and sustainable generation of useful energy, research efforts focused on the analysis and design optimization of power conversion cycles and engine technologies, such as the Stirling engine [7–9] the Atkinson cycle engine [10], and other novel engine cycle technologies [11], have also been the subject of recent publications. These studies are of interest because they have often focused not on incremental improvements to existing technologies, but rather on determining the optimal design of an engine concept based on its operating principles and scientific theory. This trend in part shows that design optimization is not just a tool for analyzing the finer details of a design in the hopes of improvement, but that design optimization can also be used as a decision-making tool at various stages in the design process. Multi-objective optimization techniques are particularly useful in this manner as they produce a set of design alternatives, usually in the form of a trade-off between competing design objectives, and these trade-off curves can be used to guide decisions about the design of new technology in various stages of the design process [12], and has been applied to energy system design problems [13]. This can be particularly helpful when developing a new technology that must immediately be cost effective when introduced to the market in order to have a chance of displacing more mature existing engine technologies.

The work presented in this paper focused on just such an effort. A novel reciprocating engine technology was developed and patented by researchers at Lawrence Livermore National Laboratory (LLNL) [14] and a prototype of that engine technology was fabricated. The inventors of the technology believe that the new engine has the chance to be competitive in the market place, but have questions about the proper sizing and configuration of the engine in terms of output power and efficiency. This paper presents the results of a model development and design optimization effort conducted for LLNL for this new technology currently under development. This study was focused on creating a computational model that predicted the performance of a specific configuration of the new engine technology based on first principles and that could produce results as quickly as possible in order to work well within a design optimization framework requiring numerous iterative evaluations of the model. In order to accomplish this task, and due to the maturity of the technology, numerous simplifying assumptions had to be made during the model development process. However, the decrease in model fidelity resulting from these simplifications was deemed to be acceptable within the context of the study, which was focused on sizing and configuration decisions based on first principles and engine theory. The models developed and the processes taken as outlined in this paper can be relevant to other research and development efforts as they demonstrate the advantages to be gained through the use of a system design models developed expressly for the purpose of implementing design optimization techniques early in the design process focused on a new technology. The results presented here do not attempt to make any claims about the capabilities of the new technology patented by Bennett et al. [14]; rather, this paper is focused on documenting the optimization focused model development process used to analyze this new technology for the purpose of making early design stage decisions.

The remainder of this paper is laid out as follows: Section 2 contains the background details on the novel engine technology studied, Sec. 3 contains the details of the engine design model developed to predict the performance of various configurations of the new engine technology, Sec. 4 contains the background and formulation of the multi-objective study conducted, Sec. 5 contains some sample results obtained, and Sec. 6 contains some brief concluding remarks. Due to intellectual property restrictions, numerous details associated the exact results obtained by this study have been omitted from this paper.

## The Harmonic Steam Engine

As previously stated, the work presented in this paper focused on the design optimization of novel engine technology for use in determining the next step to take in the development efforts associated with this new technology [14]. The engine technology has been named the “harmonic steam engine” by its inventors, and the basic function of the engine is based on the operating principles of classic reciprocating steam engines used in locomotives in the late 19th and early 20th centuries [15]. The key modification to the classic reciprocating steam engine proposed by this new technology is the use of reed valves in place of mechanically controlled intake and exhaust valves. Reed valves are thin pieces of material (usually metal) that deflect under applied forces (usually generated by fluid dynamics and/or pressure differences) to open and/or close the ports that allow steam to enter and leave the cylinders of the engine. Figure 1 below provides a cut-away schematic of a single cylinder of the harmonic steam engine technology. Note this configuration is very similar to any other reciprocating (e.g., piston-cylinder) engine with the key difference of having reed valves at the intake and exhaust ports. The exact geometry and configuration of the prototype harmonic steam engine is not provided here for intellectual property reasons, but for a point of reference the displacement of the existing prototype is on the order of approximately 165 cubic centimeters (cc) and readers are directed to Ref. [14] for more information on this invention.

Obviously, the presence of the reed valves is what drives the operation of this engine technology. Furthermore, the interaction of the engine cylinder motion and the operation of the reed valves have been observed in the prototype to be fairly complex depending of operating regime of the engine and the speed of the output (crank) shaft. However, the basic operation of the harmonic steam engine can be summed up as shown in Fig. 2.

The engine cycle begins in the top left corner of the figure with the intake valve open and the cylinder filling with steam at manifold pressure. At some point during the filling process, the intake valve will close (as shown in the top right corner of the figure), trapping a set volume of steam in the cylinder, and then the steam in the cylinder will expand in accordance with basic thermodynamic principles. The valve closing action is typically called “cut-off” in the context of steam engines [15]. Based on the design of the engine technology, at some point near the end of the expansion process, the exhaust valves of the engine will be driven open, the pressure in the cylinder will drop to exhaust pressure, and the engine will begin to reject the steam from the engine. This exhaust process continues until the piston reaches the top of cylinder and the intake valve reopens to begin the cycle again.

A notional pressure–volume (P–V) diagram for this engine cycle is provided in Fig. 3, and details on the corresponding state points listed in the figure are provided in Table 1. The basic P–V diagram for this engine is very similar to the P–V diagram for any reciprocating steam engine; however, the key issue associated with this engine technology is mechanism by which the intake and exhaust valves are opened and closed. In a traditional steam engine valve, control is only accomplished through mechanical means. Valve motion in this engine can be driven by the mechanic properties of the valves themselves (material, deflection rate, etc.), by the fluid mechanics of the steam flow past the valves, and/or by mechanical contact between the valves and other moving engine components (e.g., contact with the piston). The modeling and analysis of these key interactions represents the key challenges faced by this work.

## Engine Design Model

In this section, the details of how the operation of the engine technology presented in Sec. 2 will be presented, including all simplifying assumptions made for the purposes of computational speed as the model of the engine technology was developed based on first principles with the express purpose of being implemented within an optimization framework.

### Model Overview.

In order to effectively model the harmonic steam engine concept, it was critical to make numerous decisions about the design space of the problem. In other words, decisions had to be made about what aspects of the engine concept would be included in the model and what parameters would be disregarded. The decision was made to only model the engine as a single cylinder, leaving any multiple cylinder effects for a later study. Additionally, it was decided to limit the study to a working fluid of 100% quality saturated steam at a given temperature, and to assume that the working fluid will behave as an ideal gas throughout the engine cycle. This simplifying assumption was a rather large departure from actual engine behavior, as wet steam and condensation effects are clearly observed in the existing prototype as would be expected. That being said, the errors that would be generated by this simplification were determined to be acceptable as they would: (a) make it possible to generate a reasonably effective engine model in a short period of time, (b) would lead to a model that over estimates engine performance, thus providing a theoretical upper bound on engine performance for given configuration, and (c) could serve a starting point for higher fidelity modeling efforts that could be undertaken in future studies. Various other simplifying assumptions were made with regard to the development of the engine model, and these assumptions have been listed in Table 2. Recall that the goal of this effort was to develop a computationally efficient optimization model for conceptual design trade studies, so computational simplicity and speed were believed to be more critical than high fidelity modeling of the underlying physics.

The critical question that this study was seeking to answer through the use of an optimization study was how should this novel engine technology be configured and run in an effort to produce as much power as possible with a high level of thermal efficiency. The variable parameters of the engine system were determined to be the initial conditions of the working fluid entering the cylinder of the engine, the basic geometry of the piston/cylinder assembly, and the geometry of the intake and exhaust reed valves. All parameters of interest considered in this research are listed in Table 3 below.

Because this study was focused on computational speed, and the results produced were to be used to make conceptual design decisions, it was determined that the model should take as input the dynamic properties (spring rate and natural frequency) of the reed valves as opposed to their basic geometry (thickness, width, length, material, etc.) as this would greatly reduce the number of variable input parameters to the model while still capturing the underlying relationships between reed design and engine configuration. Clearly, it is possible to design a reed that has specific dynamic properties if those desired dynamics are known making this model simplification acceptable. It should also be noted that the prototype engine contains multiple features that physically interact with the reed valves at different times in the engine's cycle in order to override the dynamic behavior of the reeds when necessary. The exact design, shape, and mechanics of these attributes are not critical to the model developed to this work, and thus are purposefully not included in this paper in detail. However, it was critical to model the effects of those physical attributes clearly and so it was determined to include input parameters to the engine model that could drive when the valves were physically contacted as a function of crank angle position as opposed to modeling the exact size and shape of the piston head, valves, and other moving components that make these interactions possible. Table 4 contains the variable parameters of the engine model as developed, and where these parameters differ from the physical prototype parameters of interest (Table 3) have been highlighted in bold font.

### Simulation Process.

A crank-angle (*θ*) discretization strategy was chosen to simulate the operation of the harmonic steam engine, as this technique is used throughout the internal combustion engine community to model engine behavior, e.g., see Ref. [16]. This strategy involved developing a series of engine component and behavior submodels that were a function of engine crank angle, and to then link these models together within a loop that incremented the crank angle position over time (e.g., from 0 deg to 720 deg for two full iterations of the crank) and tracked the resulting engine performance. Figure 4 contains a flow chart of this process and the necessary submodels developed to support this modeling strategy. The process starts by fixing the engine configuration based on input parameters selected, and then initializes the model with values for the various model parameters (e.g., cylinder pressure, valve angle, etc.) necessary to solve various differential equations needed to determine the model parameters at the next crank angle step. The key submodels developed were concerned with the position (movement) of the piston, intake valve forces, intake valve dynamics, valve positions, and cylinder pressure. Once the engine model has evaluated all submodels for all crank angle positions of interest, a final submodel would analyze the produced model parameters as a function of crank angle (e.g., work, torque, etc.) and produce values for the output work, power and efficiency of the engine for a given set of input parameters.

### Piston Movement Model.

Recalling the general governing assumptions stated earlier in Table 2 (e.g., no friction and thermal losses), the following calculations were used at each crank angle position to determine the position and speed of the piston, the forces generated and the resulting output power and torque of the engine cylinder.

*ω*the velocity of the piston was determined using Eq. (2)

*φ*) changes with crank angle, it was important to track that angle using Eq. (5). Given the instantaneous force generated by the piston and the connecting rod angle the torque generated by the piston on the crank shaft by piston as a function of

*θ*can be calculated using the following equation:

### Intake Valve Model.

As seen in Sec. 3.3, the pressure in the cylinder was a key parameter for determining the output torque of the cylinder (and the resulting output power of the engine). The key to knowing the pressure in the cylinder is knowledge of the position of the intake valve. Referring to Fig. 3, if the intake valve is open, the cylinder is assumed to be at manifold pressure, and that if the exhaust valve is open for any reason, the cylinder is assumed to be at exhaust pressure (zero gauge). The intake valve is the most critical element of this system, as when it opens and closes, it has a direct effect on the power produced and the thermal efficiency of the engine based on basic reciprocating steam engine theory [15]. A close-up view of the intake reed valve is provided in Fig. 5.

*θ*) as a function of engine crank angle (

_{v}*θ*) was of primary interest. The motion of the intake valve was modeled with the second-order ordinary differential equation provided in Eq. (7). Based on the assumed reed geometry and standard cantilevered beam theory, the damping coefficient and respective terms were assumed negligible in comparison to the mass and spring constants of the reed. Since the reed dynamics as opposed to reed geometry were the focus of this model, the mass of the reed had to be back calculated as function of spring constant and reed natural frequency as provided in Eq. (8)

*θ*

_{v,i+}_{1}) for recursive output. A Jacobian transformation was utilized to map system level variables (e.g.,

*θ*) to the variables local to the valve dynamics (e.g.,

*y*). A simplified form of the resulting numerical differential equation is shown in the following equation:

_{v}*θ*= 0), the force applied to the valve was calculated using Eq. (13)

_{v}When the valve was determined to be open, the force applied was calculated using fluid dynamic analysis. As detailed in Ref [17], two models were used to calculate the applied aerodynamic force to the intake valve, a computational fluid dynamics (CFD) based solution and an analytic solution. The CFD solution was calculated within the model using a kriging surrogate approximate model, fit to a discrete set of CFD solutions generated via a space-filling design of experiments technique that varied the inputs to the CFD model. In other words, a discrete set of CFD models were built and evaluated with different parameter values (i.e., intake pressure, valve angle, port area, etc.) and the resulting valve force value produced was recorded. Using the discrete input/output CFD data, the kriging curve-fitting technique was then used within the model to predict valve force values for untested combinations of intake valve variables. For details on the kriging techniques, see Ref. [18]. An example CFD solution is provided in Fig. 6 below.

The kriging approximation technique has the unique property of not only producing an approximate value of the output function of interest, but also producing a predicted error value for that approximate value [18]. When the predicted error value for the approximate CFD output produced by the kriging model was determined to be too high, the valve force was calculated within the model using an analytical Navier–Stokes solution for a simplified convergent-divergent nozzle model, which was determined to be an acceptable simplification of the actual valve port geometry flow scenario. Figures 7 and 8 show the actual flow scenario of interest and the corresponding convergent–divergent nozzle model used to approximate the flow. Clearly, in this situation, the inlet and throat areas are calculated based on the port area and the valve angle. As stated previously, the details of this flow solution technique can be found in Ref. [17].

### Cylinder Pressure Model.

In order to determine the amount of work produced by the piston during the engine operation, a pressure model was developed. Based on the stated assumptions used to develop the engine model as outlined in Table 2, the cylinder pressure was determined in three distinct ways for each given crank angle position based on the position of all intake and exhaust valves, which, as previously stated, could be determined based on engine input parameter values or the intake valve dynamic model described in Sec. 3.4.

*Option 1*: If the intake valve was determined to be open, the cylinder pressure was assumed to be instantaneously equal to the manifold pressure, reduced by an assumed volumetric efficiency of 70 percent.

*Option 2*: If the exhaust valve was determined to be open, the cylinder pressure was assumed to be instantaneously equal to the exhaust pressure (zero gauge).

*Option 3*: If both the intake and exhaust valves were determined to be closed, then the cylinder pressure was determined to be slightly lower than the cylinder pressure at the previous crank angle position assuming an adiabatic polytropic expansion process as depicted in Eq. (14).

### Engine Output Model.

*θ*) positions of interest, the engine output parameters for the entire engine cycle can be evaluated. The output power of the engine was calculated using Eq. (15), which is a function of the engine torque that was calculated for each

*θ*value using Eq. (6) and the process described in Sec. 3.2

*ω*. As a result, in many cases, the more relevant output value would be the average cycle torque, which was determined by calculating the average engine torque values for all recorded

*θ*values. The other output parameter determined by the engine model is the cycle efficiency, which was calculated using Eq. (16). The efficiency was calculated as the ratio of the total energy produced by an engine cycle (complete rotation of the crank shaft) as provided in Eq. (17), divided by the internal energy of the steam in the cylinder at the time the intake valve closes (i.e., cut-off) as provided in Eq. (18)

## Optimization Study

As previously stated, the overall goal of this work was to use design optimization to study the harmonic steam engine technology in attempt to answer sizing and scaling questions to assist in the conceptual design of this novel technology. This section will present some brief background details on design optimization and the specific harmonic steam engine design optimization problem solved in this study.

### Design Optimization.

**, to an analytical, numerical, or computational model that maximizes or minimizes one or more of the model outputs,**

*x***, provided a set of constraint values,**

*f***, are satisfied, while also considering a set of fixed input parameter values,**

*g***. The mathematical formulation of the design optimization problem is formulated in the following equation:**

*p*Figure 9 shows a general “black box” model that represents the structure of a design model suitable for design optimization. Optimizing the design of a system involves developing a computational framework where the inputs to a black box model can be varied systematically and the resulting ** f** and

**values can be determined and evaluated. The model is described as a black box because the exact nature of how the variables,**

*g***, and parameters,**

*x***, are mapped to the objectives,**

*p***, and constraints,**

*f***, are not important to how the design optimization problem is formulated and solved. There are many excellent design optimization techniques and numerous computational tools are available for implementing design optimization on various types of system design models. Please see Refs. [19–21] for more details.**

*g*### Multi-Objective Design Optimization.

Often, design problems will have multiple objectives, or performance measures, that are of interests to the overall design process. In those cases, multi-objective optimization techniques are more appropriate. Multi-objective optimization techniques are well studied [20,21] and have the unique property of producing a set of optimal solutions for a design decision-maker to consider. In contrast, many optimization studies in the literature (e.g., see Ref. [12]) use traditional optimization procedures that only produce a single optimal solution. The set of solutions produced by multi-objective solvers is called the Pareto frontier and essentially represents a trade-off between the multiple competing design objectives of interest. Figure 10 shows the relationship between a set of solution in the decision space (design variables values) and the resulting optimal Pareto frontier in the objective space (mapped objective function values). Multi-objective design optimization problems can be solved with various techniques, but the work presented in this paper used the multi-objective genetic algorithm technique, which is a population-based meta-heuristic solver [21].

### Steam Engine Optimization Problem.

*η*and

*T*, it was necessary to minimize the negative of both of these objectives, which is a common practice in design optimization (see Ref. [19]). Figure 11 depicts the implementation of the harmonic steam engine model detailed in Sec. 3 as a black box optimization model. In this study, the harmonic steam engine model was implemented in Matlab and Eq. (20) was solved using Matlab's build-in multi-objective genetic algorithm solver (gamultiobj.m) with a population size of 100, 100 generations, and all other solver parameters set to their default values

The design variables implemented in this optimization study, including the upper and lower bounds on those variables, are provided in Table 5. All other input parameters to the harmonic steam engine model were set to fixed values in the optimization study and are thus considered as design parameters, ** p**.

Note that in Eq. (20) and Fig. 11, the constraint functions of this optimization problem are listed as “Flags.” The constraints of the harmonic steam engine were a set of variables that were initialized with a value of zero within the harmonic steam engine model and could be switched to have a value of one if any of conditions were observed within the model that were beyond the limits of the capabilities of the model. These flags implemented within the harmonic steam engine model are listed in Table 6, and if any of these conditions are observed within a particular execution of the harmonic steam engine model, the optimization solver then considers that particular set of input variables (e.g., ** x**) to be infeasible and thus disregards that particular solution.

## Results and Discussion

### Harmonic Steam Engine Results.

The harmonic steam engine model was developed and tested with various input parameters to ensure that the model performed well within the limits of the stated assumptions, could effectively predict engine performance over multiple full rotations of the engine's crank shaft, and could produce results that reflected the trends observed in the prototype engine. The crank-angle discretization step size could be varied as desired by the user, but typically a step size of between 1 deg and 5 deg was observed to produce the best performance. For the final results presented in this paper, a crank angle step of 1 deg was utilized and four complete rotations of the crank shaft were modeled. All fixed parameters of the engine model were set to the values observed in the existing prototype engine, but the exact values have been omitted from this paper for intellectual property reasons.

The model performed well and was observed to be able to complete a full simulation of the engine in 5–10 s on average. Figures 12 and 13 contain output data from a single simulation of the engine model, showing the predicted P–V diagram and the predicted valve movements of the engine. Note that the P–V diagram produced by the harmonic steam engine model (Fig. 12) has the correct shape and trends as outlined in Sec. 2. Also, the valve movement predicted in Fig. 13 correctly matches the pressure changes observed on the P–V diagram, indicating the simulation model was effective at modeling the physics of the engine system within the limits of the simplifying assumptions stated in Table 2 (e.g., adiabatic expansion, isentropic processes, etc.).

### Optimization Study Results.

In order to produce useful sizing and scaling trade-off data to harmonic steam engine design decision-makers using the developed engine simulation model, a series of multi-objective optimization studies were performed by solving Eq. (20) with varying sets of input variables considered. The first study involved using the optimizer to vary the values for the first seven parameters listed in Table 5 (e.g., *x*(1)–*x*(7)), while holding *x*(8) fixed to the value of the bore used in the existing stock prototype engine. Next, three optimization studies were performed with fixed cylinder bore values of 85 mm, 100 mm, and 115 mm, respectively, all of which are larger than the bore of the stock prototype engine. In other words, the optimizer varied *x*(1)–*x*(7) again, while *x*(8) was fixed to values 85, 100, and then 120 mm in the next three successive optimization studies to determine the potential effect of cylinder bore on engine performance. In each of these studies, the stroke of the engine was determined using a fixed bore to stroke ratio of 1.2, which was determined to be optimal based on a review of internal combustion engine research studies focused on optimal volumetric efficiency (e.g., see Ref. [22]). The fifth and final study considered all eight variables listed in Table 5 (e.g., *x*(1)–*x*(8)), and used a fixed bore to stroke ratio of 1.2. The Pareto frontier results produced by each of these studies are all plotted together in Fig. 14, along with the efficiency and average torque the harmonic steam engine model predicted for the prototype engine geometry for comparison purposes. The design variables and fixed parameters considered in each of these studies are summarized in Table 7 below.

As can be seen in Fig. 14, the harmonic steam engine model predicts a strong dependency between cylinder bore and engine torque, and furthermore the results show that it is theoretically possible to get improved efficiency in the engine through effective selection of the other engine parameters considered, particularly the variables that determine the dynamics of the intake reed valve. The absolute values of the torque and efficiency predicted have been omitted for intellectual property reasons, but the results clearly show the power of using a predictive model and multi-objective design optimization to assist in the conceptual design of a new engine technology as marked improvement over the current prototype engine is predicted to be possible.

Another product of using multi-objective design optimization to assist in the conceptual design process of a complicated system such as the harmonic steam engine is that the results can highlight the correlation between the optimal values for system parameters of interest. A sample of this is provided in Fig. 15, which depicts scatter plots of all of the obtained multi-objective optimal values for 4 of the model input parameter in comparison to the predicted optimal objective (e.g., *η* and *T*). The correlation coefficients for each of the parameter relationships were calculated and plotted on the figure, and parameter relationships that have a significant correlation are highlighted with black markers. As an example of the types of information that can be determined from the optimization data, this plot shows that along the determined Pareto optimal frontiers for the harmonic steam engine a strong correlation between the natural frequency of the intake reed valve was predicted.

## Conclusion

In this paper, the details of a research effort focused on the development of a system-level computational model of a novel steam engine was developed from first principles for the express purpose of performing design optimization studies of the engine technology.

Various modeling decisions were made in an effort to balance desired model accuracy with necessary computational speed, and the resulting computational model was determined to perform effectively for making conceptual design decisions within the limits of the assumptions made. The results produced by the model and the optimization study correlated well with the known performance of the existing prototype harmonic steam engine and provided engine design decision-makers with a variety of information that can be utilized to make sizing and scaling decisions as the technology matures and is further developed.

## Acknowledgment

The work presented here was partially supported by Lawrence Livermore National Laboratory contract B604738, a subcontract of Department of Energy contract DE-AC52-07NA27344. The authors would also like to thank the department of Mechanical and Aerospace Engineering Department at California State University Long Beach, where this research was initiated. An early version of this paper was presented at the 2014 ASME International Mechanical Engineering Congress & Exposition [23].

## Funding Data

Lawrence Livermore National Laboratory (Contract No. B604738).

U.S. Department of Energy (Contract No. DE-AC52-07NA27344).

## Nomenclature

*A*=area

*A*_{eff}=effective area

*A*=_{p}port area

*B*=cylinder bore

- BDC =
bottom dead center

*c*=_{p}specific heat capacity

*E*=energy

*f*=objective function

*F*=force

*F*=_{v}valve force

*g*=constraint function

*h*=clearance height

*k*=stiffness

*l*=connecting rod length

*L*=stroke

*l*=_{v}valve length

*m*=total rotating engine mass

*m*=_{v}valve mass

*p*=design parameter

*P*=pressure

*R*=crankshaft radius

*t*=time

*T*=torque

- TDC =
top dead center

*U*=internal energy

*v*=velocity

*V*=volume

*x*=design variable

*y*=position

*y*=_{v}valve position

*γ*=specific heat ratio

*η*=efficiency

*θ*=crank angle

*θv*=valve angle

*ρ*=correlation coefficient

*φ*=connecting rod angle

*ω*=rotational speed

*ω*=_{n}natural frequency