Using Mechanical Event Simulation in the Design Process

ABSTRACT

In this paper, we show how to incorporate Mechanical Event Simulation (MES) into the design process of mechanical components. MES relies heavily on nonlinear Finite Element Analysis (FEA). Our goal is to enhance the design process by considering the event that the part in question will undergo. Specifically, we aim to utilize virtual experiments on the computer to verify the validity of engineering designs. These experiments should simulate the interaction of the part with its surroundings.

Typical engineering designs rely on isolating the part from its surroundings by introducing boundary loads and conditions. These boundary constraints are relatively easy to determine for parts that experience only static conditions. Linear FEA has proven to be a valid tool in the design of such "static" components. For problems involving motion, engineers generally have also relied on linear FEA, but have had to both (i) utilize rule-of-thumb methods to estimate the magnitude of these loads, and (ii) artificially constrain the location of the part in order to ensure that the linear limit is not exceeded. The effects of a drop test represent a typical problem in which linear FEA has been utilized to analyze a dynamic event. By embracing MES, engineers are freed from having to make such compromises.

MES relies heavily on nonlinear FEA which has been regarded as notorious for its instabilities and computational inefficiencies. These drawbacks to nonlinear FEA have, in the past, greatly limited its use. Because of our ambitious goal to include nonlinear FEA in the design of a wide range of mechanical components, we must mitigate these drawbacks.

In this paper, we present results obtained with a nonlinear FEA processor whose state-of-the art adaptive solution method and fully automatic time-stepping scheme remove the stability concerns. These state-of-the-art techniques combined with the introduction of a new type of element, which we refer to as “kinematic," significantly reduce solution times. This robust and efficient processor is the first of our three-part approach to using MES to guide designs. Secondly, because we are particularly interested in dynamic events, the processor must be able to simulate large-scale motion and its consequences, from parts making contact to their fracture. This brings us to our third and final part: to closely duplicate material behavior. Besides fracture and the standard material models, we have added material-based damping. This damping allows us to consider real-world problems where such dissipation is always present.

The types of real-world problems that we can solve are extensive. As an example, we consider the design of a tank. We focus on the validity of the design if we demand that the tank's structural integrity withstand a high-speed collision. Our processor lets the engineer simulate the entire virtual event during which the intact tank collides and possibly even ruptures. From the results of this virtual experiment, the engineer can easily determine the adequacy of the design. Thus, the engineer can focus on the physics of the problem, rather than concentrating on modeling approximations such as the ad hoc estimation of critical loads.

1. INTRODUCTION

MES represents a paradigm shift in the design of mechanical components. It allows engineers and designers to simulate the actual conditions that a mechanical component will experience; that is, the event associated with its application. MES differs significantly from the general practice of using linear static FEA in the design process. Linear static FEA is the culmination of methods introduced by Galileo and da Vinci [1]. Even though it is well understood that linear static analysis has limited applicability, such methods are still an integral part of an engineering education. Linear methods are even used to model motion using pseudo-dynamic analyses that completely neglect inertia. MES is a true dynamic analysis tool because it considers inertia and is not based upon a linear static perspective.

The use of linear static FEA is generally rationalized by considering harsher than expected conditions. This is accomplished by using safety factors in conjunction with engineering estimates of the mechanical loads. When the component in question will experience only static conditions, linear static FEA may suffice as an analysis tool.

Of course, most static situations are the result of an event that included motion. MES is geared toward the design of components that experience motion, from the event that precedes a static situation to continuously moving operations. When motion exists, not only are the loads no longer constant, but their magnitude and direction are not easily estimated. MES completely bypasses the need to estimate these loads. That is because MES accounts for both (i) the interaction of the component with its surroundings, and (ii) the inertial forces generated by the motion of the component itself. Before describing how MES uses nonlinear FEA, it is important to show how even conservative estimates of loads in the design of moving components can result in severe under-prediction of mechanical stresses.

Dropping a nearly rigid object onto a flexible, simply supported beam clearly demonstrates how difficult it is to estimate loads when motion is involved. We are simply interested in obtaining a value for the maximum value of the stresses within the beam. The estimate that we need is the maximum impact force that the falling object has on the beam. Classical beam theory can then be employed to yield the maximum stress. Of course, classical beam theory is only appropriate under static conditions, but we expect the stresses within the beam to be much more dependent on displacement (deformation) than on the details of the motion. Specifically, we assume that the stresses are insignificantly affected by the inertia of the beam. This assumption will be revisited later in this section when we discuss the results of an MES of this problem.

The analysis is based on both energy conservation principles as well as on classical beam theory; a more detailed description of the analysis can be found in [2]. In order to obtain a closed-form analytical solution, we consider a problem with a simple geometry .The problem consists of dropping a small 4.0 lb. weight, W, from a height, H, of 1.0 in. onto the center of a simply supported steel (Young's modulus, E, of 30 x 10⁶) beam with a length, L, of 23 in. and a circular cross-section with a diameter, d, of 1/2 in. The weight is assumed to make contact only along a plane at the middle of the beam and perpendicular to its primary axis. Finally, without sacrificing the accuracy of the results, the dropping weight is assumed to be rigid. In order to further verify our results, we also conduct a physical experiment of this problem.

We first pursue the classical physics solution for the maximum impact force. A simple energy balance between two end states can be used to obtain the maximum displacement of the beam. The initial state is defined by the weight, W, at rest at height H above the beam. The final state consists of the beam at its highest deflection and the weight having no velocity - the point at which the velocity of the weight changes direction. This energy balance gives the following expression for the maximum displacement of the center-point of the beam

A Mechanical Event Simulation (MES) of the same problem was conducted. This provides for a numerical verification of the analytical solution and vice versa. Under this type of analysis the assumptions inherent to classical beam theory are not necessary. Thus, the beam can possess mass and, hence, inertia. Furthermore, the weight is not modeled to be rigid; instead it is assumed to be made of steel (E of 30 x 10⁶ lb./in.²). Contact elements were utilized to model the impact between the mass and the beam. Details regarding, contact elements are discussed in section 2.

The MES yields a maximum force of 51.7 lb. and stress of 24,700 lb./in.². Note how the analytical solution yields slightly higher values for both the maximum force and stress. We hypothesize that the primary reason for the discrepancy is that the analytical solution assumes that the beam was massless. In the MES, as in reality, the beam possesses mass. This mass combined with the motion of the beam generates inertia, which absorbs some of the impact energy. We verified our hypothesis by performing a second MES in which we assumed the beam to be massless - an unphysical, yet mathematically valid assumption. This latter MES yielded a maximum impact force of 57.6 lb., which is within 1% of the analytical result. The physical experiment of the problem yielded a maximum impact force greater than 50 lb. Even though all three methods used to obtain the maximum force yielded slightly different results, we are confident that the short drop did generate at least 50 lb. of force.

The example just discussed certainly demonstrates how the design of a component that will experience motion requires special handling. This example has been configured so that analytical methods would be applicable. Real-world problems are rarely that simple, and thus require numerical analysis. One could opt for linear FEA combined with significant loading related estimates, or one could choose MES. The latter choice does require a larger computational effort, but does mitigate the number of engineering man-hours spent performing estimates and assumptions. In the following sections of this paper, we discuss how we have enhanced nonlinear FEA to the level that a typical designer can use MES. The enhancement has required that the nonlinear FEA processor "underneath" MES be efficient, robust and able to simulate a wide range of mechanical scenarios. We first discuss how the processor is robust. Then we tie this robustness to how MES is efficient from both a user-intervention as well as from a computational perspective. The use of MES not only demands the simulation of motion, but also of its aftereffects. We discuss two such effects: impacts and accurate descriptions of material behavior. Finally, we discuss how MES is used to design an actual mechanical component.

2. MES SOLUTION METHOD

MES is heavily based on nonlinear FEA, which has generally been regarded as both highly unstable as well as computationally intensive. Both of these drawbacks have been addressed in the development of MES. The instabilities inherent to nonlinear FEA can usually be eliminated using a smaller time-step size, t. Of course, the smaller t, the larger the computing time. This trade-off between stability and efficiency can be significantly diminished by incorporating an "intelligent" (automatic) algorithm to change t throughout an event.

MES utilizes such an algorithm; t is automatically reduced during critical times and again increased if/when conditions no longer warrant such temporal detail.

The heart of the algorithm is the method used to identify "critical times." We use the rate of convergence of the nonlinear iterations to gauge stability. Such iterations are required to achieve mechanical equilibrium at every time-step. The rate of convergence is determined by examining how the iterative residual changes from one iteration to the next. During "critical" periods this residual, will typically not experience a monotonic decline. Conversely, when the residual experiences monotonic declines, critical events are rarely occurring. These two types of behavior by the residual can be used to formulate criteria to trigger t decreases and increases. It should be mentioned that the scheme does not just reduce t at critical times, but it backtracks in time in order to ensure that the entire critical period is captured. Finally, t is also decreased during highly critical points at which not even one iteration is obtainable. Such highly critical points (i.e., the onset of buckling) generally require successive t reductions until a stable solution is obtained.

The automatic time-stepping scheme does have the added consequence of making MES efficient from the perspective of total analysis time. Specifically, the scheme diminishes user intervention by removing the common practice in nonlinear FEA to manually and intelligently change t throughout an analysis. Nevertheless, we still have to contend with the large computational efforts usually required by nonlinear FEA. After all, in order to achieve mechanical equilibrium, each time-step requires the solution of at least one equivalent linear FEA problem. Nonlinear mechanical equilibrium is achieved using Newton-Raphson iterative methods that generally require the solution of several linearized forms of the nonlinear problem [3].

In order to diminish run-times, MES includes a new type of continuum finite element. This element type, which we refer to as "kinematic," does not experience strains, and thus does not report stresses. Otherwise, kinematic elements behave just like their equivalent 2 or 3-D flexible counterparts. That is, they can have mass, have loads applied on their nodes and/or faces, and, more importantly, experience motion. Their advantage over their flexible counterparts is that they barely contribute to the size of the global stiffness matrix; thus their use can greatly improve run-times. Kinematic elements are particularly applicable in models with an appreciable number of elements known to experience insignificant deformations, but large scale motion, during an event. Such prior knowledge of a model's behavior does require some engineering expertise or prior experience with the model. Designers generally possess this knowledge. Hence, kinematic elements can be an integral part of using MES in the design of moving parts.

Using MES to design moving parts requires that the underlying nonlinear FEA processor be able to handle the consequences of motion. In this section, we concentrate on contact/impact. In the next section we discuss other possible consequences related to material behavior. MES supports two types of contact/impact: (i) surface-to-surface, and (ii) surface-to-rigid plane. Surface-to-surface interactions are accomplished using contact elements. These elements allow for any surface of any solid (including 2-D representations) to interact with any other such surface (including itself) during an event. Actually, it is even possible to begin an event with surfaces already in contact. This surface-to-surface interaction is particularly useful when designing mechanical assemblies. The surface-to-rigid plane contact provides for a highly computationally efficient means to simulate objects impacting and/or sliding on flat surfaces. The MES of a standard drop test represents an ideal application of this form of contact. It should be mentioned that both forms of contact are formulated using energy conservation principles.

3. NEW MATERIAL MODELS FOR MES

A further consequence of requiring an accurate simulation of motion is proper material modeling. Besides the material models generally handled by nonlinear FEA, we have focused on three specific kinds of material behavior: (i) detailed stress-strain curve with yielding, (ii) fracture triggered at threshold stress level, and (iii) mechanical dissipation through stiffness-dependent damping.

Case (i) is a generalized, curve description, form of the standard (bilinear) von Mises material model [4]; the curve portion is for strain levels above yield and is represented using a sequence of line segments (see Figure 1). This added detail makes the curve description model better capture material behavior beyond the yield point.

Case (ii) consists of allowing elements to lose their stiffness once a user-specified tensile and/or compressive breaking stress level is reached for a given material. This procedure is relatively simple to implement, but the sudden nature of such breaking/fracture does require the robust solver discussed in section 2. This ability to have elements "break" has allowed us to model the fracture of brittle materials. Figure 2 shows the results of an MES of shattering glass.

Case (iii) is an element-based form of Rayleigh damping [5] used to model mechanical dissipation within individual elements. Specifically, we aim to model the mechanical energy losses incurred solely because of material deformation rates without considering heat transfer details. Including such details is generally unnecessary and would significantly complicate as well as lengthen the duration of an analysis. Instead we chose to adapt standard damping theory to simulate the loss of mechanical energy into thermal. We refer to this form of energy loss as material-based damping. This damping is modeled as proportional to an element's strain rate and to its stiffness matrix, but not to its mass matrix. Making the damping proportional to the latter matrix would have resulted in the damping of free motion - which is not how physical dissipation manifests itself. To utilize this material-based damping requires the specification of a multiplier. This multiplier is analogous to the stiffness-dependent parameter in Rayleigh damping. We take the multiplier to be a constant independent of operating conditions and to depend solely on the type of material. Thus, making this constant a material property.

Figure 2: Results of an MES of a moving object shattering a plane of glass.

This material constant can easily be determined using a cantilever beam arrangement. Consider a beam mounted vertically on a floor; the primary axis of the beam should coincide with the direction of gravity. A mass should be affixed on the free end of the beam. The experiment consists of (i) manually displacing the free end of the beam a small distance perpendicular to the primary axis, and (ii) then letting the beam oscillate freely while measuring the displacement of the mass. To obtain the constant, several MESs of the experiment are conducted; each MES having a distinct value for the material constant for the beam's material. The MES whose value best matches the experimental data provides the value for the constant. The results of three such MESs are shown in Figure 3. Each MES uses an aluminum alloy 2024-T4 (with Young's modulus of 10.9 x 10⁶ lb./in.², poisson's ratio of 0.397, and specific weight of 0.101 lb./in.³) rectangular beam with dimension 20 by 1 1/2 by 1/4 in. The manual displacement was simulated using prescribed displacements in the direction of the minimum dimension of the beam. From Figure 3 one can observe the symmetric form of each time-trace. If the beam had been affixed horizontally, gravity would have caused an asymmetry - an unnecessary complication. It should be noted that because our material damping model does not include dependence upon operating conditions, the value obtained for the constant is most appropriate when simulating events under similar conditions (i.e., temperature) to that of the experiment.

4. USING MES IN THE DESIGN PROCESS

In the presentation corresponding to this paper, we discuss how MES can be used to design a tank. We focus on designing the tank such that it can withstand a high-speed collision when full with liquid. The collision is modeled using the surface-to-rigid plane contact discussed in section 2. The kinematic elements also discussed in section 2 are used only in portions of the tank known to experience insignificant deformations during the impact. These portions were identified during the design process by considering simpler versions of the model. The remaining portion of the tank is modeled using flexible elements with the curve description von Mises material model. As mentioned in section 3, this material model accurately describes post-yield behavior. The collision will certainly result in significant yielding, but the tank is designed to maintain its structural integrity. Specifically, care is taken so that ultimate (breaking) stresses are not reached. Thus, the MES does not need to account for the fracture of the material. Of course, we could consider an MES where fracture does occur, and observe how the tank ruptures. This would be a possibly dangerous physical experiment, but certainly a very safe virtual experiment.

The liquid is represented using hydrodynamic elements. These elements allow for the simulation of the interaction of fluids with solids without considering the details of the flow. Such interaction is typically done using hydrostatic pressure loads; hydrodynamic elements provide for a much more accurate representation of fluid-solid interaction because they account for the inertia of the moving fluid.

This design clearly demonstrates how appropriate it is to incorporate MES into the design process. There is no need to estimate loads; all that is needed is a description of the operating conditions required by the design. So instead of spending valuable time making engineering approximations, the designer performs virtual experiments of the part "in action" using MES to gauge the validity of the design.

REFERENCES

1. Timoshenko, S P - History of Strength of Materials, Dover Publications, Inc., New York, NY, 1983.
2. A Finite Element Method for Problems Involving Motion, Algor Design World, Second Quarter 1998 (issue), 1998.
3. Spyrakos, C and Raftoyiannis, J - Linear and Nonlinear Finite Element Analysis. Algor, Inc., Publishing Division, Pittsburgh, PA, 1997.
4. Malvern, L E - Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1969.
5. Bathe, K J - Finite Element Procedures in Engineering Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ, 1982, pp. 528.