A co-simulation method for system-level simulation of fluid–structure couplings in hydraulic percussion units

The TLM method was applied to modelling and simulation of distributed physical systems in [23], where it was referred to as bi-lateral delay line modelling. In short, the method is based on the one-dimensional wave equation and it introduces time-discrete model elements, TLM elements, that effectively decouple different parts of the model. This allows each separated part to be simulated individually, because they only depend on delayed values from the other side of the elements. The time delays ensure that the interface points will be weakly coupled and the method, therefore, makes it possible to solve systems using explicit methods rather than needing implicit solvers for large equation systems. In a simulation, the shortest possible delay is the same as the simulation time step \(T_{s}\). Longer delays can be realised by delaying values multiple time steps. The number of steps to use depends on the physical properties of the object being modelled, primarily its length and wave propagation speed.

The TLM method has also been used in [24] where an efficient meta-model-based co-simulation framework for mechanical systems is presented. The focus of the framework is on co-simulation of axis and ball-bearing joint applications, where both translational and rotational mechanic TLM couplings are defined. Their idea is to simulate each part of a system model in the tool that is the most suitable, but to use TLM elements at the interface points to decouple the different environments. Care is taken to insert the elements where natural physical time delays exist between the parts. Another example is [25] where the method was used in co-simulation of a complete wheel-loader. Matlab/Simulink was used to model the diesel-engine and transmission, Adams MBS for the tires, frame and mechanical load and Hopsan, for the hydraulic control system. A similar work connecting Hopsan to Adams for simulation of a forester machine is found in [26]. Here, the co-simulation interface is also realised by the FMI interface, and Adams is administering the simulation process. The hydraulic system is modelled in Hopsan and exported as an FMU that is imported directly into Adams. Further, a comparison between different co-simulation integration methods was made in [27]. The conclusion was that stability could be improved if the simulation models to be coupled were divided into two groups, simulated after one another. The TLM method was one of the methods compared, and TLM elements made up one of these groups. This method proved to have superior stability properties, but a model approximation, a so-called parasitic inductance, that depends on the length of the simulation time step was introduced. When choosing the simulation time step, it is important to be able to quantify this approximation. An example of how to calculate the relevant parasitic inductance for the TLM elements used is covered later in this section.

For the co-simulation example in this paper, loss-less, single-time-step TLM elements are used. It is assumed that the components only undergo small deformation and that linear-elastic material properties can be used. Internal friction is also neglected in the elements, but it is possible to lump frictions contributions to the boundaries [28] if they have a meaningful effect. The loss-less bi-lateral delay element is illustrated in Fig. 4. The equations describing it, derived for the time-domain [29], are given in Eqs. 1, 2 and 3 where \(p_{1}\) and \(p_{2}\) are the “effort” variables and \(Q_{1}\) and \(Q_{2}\) are the “flow” variables at the respective boundary. During the delay duration, compression (fluid) or strain/stress (solid) waves represented by \(c_{1}\) and \(c_{2}\) travel in the respective directions through the element. The loss-less elements characteristic impedance \(Z_\mathrm{c}\) is a scalar property that relates the physical properties: capacitance C and inductance L, to the introduced time delay T. As is shown by the last term in Eq. 1, the characteristic impedance acts as a damping together with the boundary velocities. This fact can be used in co-simulation if the simulation method on the receiving end supports damping control. A typical example is in MBS where spring-mass-damper types of sub-models are common, see e.g. [26]. Directly affecting the damping was, however, not possible in the LS-DYNA implementation presented here, and an approximation had to be made. To calculate the pressure acting on the piston, prior to the FE-simulation step, the damping term was based on the previous flow value according to Eq. 4. For short time steps, the difference should not be that large, but this is nevertheless a possible source of error that has not been investigated further in this paper.Material or fluid properties determine C and L, while T equals the simulation time step for a single-step delay element. The fact that \(Z_{c}, T, L\) and C all depend on each other means that a choice must be made whether to model a TLM element so that the physical inductance, capacitance or characteristic impedance is preserved. In the hydraulic cavities, there is a small amount of fluid that does not move much; the dynamics is mainly affected by the fluid elasticity. The large inductive contribution comes from moving the relatively heavy piston. In that case, modelling the correct inductance (mass) would be important to preserve the dynamic response. For stationary parts, the capacitance (stiffness) is of more interest. The cavities are, therefore, modelled as purely capacitive, elastic fluid springs.

In a hydraulic percussion unit, the typical material properties of interest are those of the fluid and solid elements (piston). For the hydraulic TLM element, used as the co-simulation interface on the fluid side in this work, the Purely Capacitive model \(Z_{c}^\mathrm{PC}\) is given by Eq. 5, where K is the bulk modulus and V the volume of the fluid cavity. To cancel out the inductance, Eqs. 2 and 3 are combined. The resulting model will then preserve the desired capacitance (stiffness) regardless of the time delay T. For comparison, the characteristic impedance for a capacitive element representing the piston is given in Eq. 6. In this model, linear elasticity and a 1D simplification (no lateral expansion) is assumed. Material and component properties of interest are: initial cross-sectional area \(A_{0}\), initial length \(X_{0}\) and Young’s modulus E. In the mechanical case, capacitance is the same as 1/k, where k is equal to stiffness, and the inductance is the same as the mass. It is rarely possible to select T and set \(Z_{c}\) so that L and C perfectly coincide with real physical values. The delay T in each TLM element will cause an individual modelling error called parasitic inductance or parasitic capacitance. The ideal situation in TLM simulation is when T corresponds to \(T_\mathrm{physical}\), the real physical wave propagation time-delay of the modelled component. As a comparison, using the values from Table 2 with steel for the piston and oil for the fluid, the wave speed is \(\sim\)5172 and \(\sim\)1341 m/s, respectively. For a single-step line, one meter long, the optimal simulation time step would be \(T_\mathrm{opt,piston} \approx 1.9 \times 10^{-4}\) s and \(T_\mathrm{opt,fluid} \approx 7.5 \times 10^{-4}\) s. As the computer simulation is discrete by nature, the modelled delay T depends on the chosen simulation time step \(T_{s}\), and the number of delay steps in the line, \(T = nT_{s}\). The wave speed through the line (the speed of sound), together with the time step used, results in a perceived length of the line model. When this length is too long, then additional parasitic inductance (extra mass) will be added. At the same time, parasitic capacitance will cause the modelled stiffness to decrease. For the purely capacitive model of the hydraulic cavities, all model approximation will show up as a parasitic inductance according to Eq. 7. To represent the hydraulic inductance as its mechanical counterpart mass with unit kg, it is multiplied with the affected area in square, according to Eq. 8. A comparison of the model approximation (parasitic mass) for different time steps for a fluid cavity with volume \(2 \times 10^{-4}~\mathrm {m}^{3}\), affecting an area \(8 \times 10^{-4}~\mathrm {m}^2\) and a piston with diameter 0.07 m and length 0.4 m, is given in Table 1. The size parameters are relevant to the application example and the material properties are listed in Table 2. Both the fluid cavity and the piston are modelled as purely capacitive, and thus any time delay larger then zero will give parasitic inductance. Due to the small volume of the cavity, the time step should be kept below \(10^{-4}\) s to avoid adding noticeable parasitic mass in relation to the piston weight. The piston has a mass in reality, and there is a specific simulation time step when the parasitic mass coincides with the actual mass (12.08 kg). It is clear that different TLM element models have different optimal simulation time steps. In general, the user must choose the lowest required time step that give acceptable approximations and then use multi-step delay models for the longer components. The co-simulation interface will not handle this automatically.

Another source of approximation is that the cavity volumes have been modelled as fixed. In reality, the volumes will change when the piston moves. Equation 7 shows that the smaller the volume, the larger the parasitic inductance will become. In this case, the approximate volume of the cavities lies in the range \([1.3 \times 10^{-4}, 2.5 \times 10^{-4}]\) m\(^{3}\) which at a time step \(10^{-4}\) s would represent a parasitic mass range of [0.04, 0.08] kg. At this and shorter time steps, the difference is negligible.

Note that the piston model in Eq. 6 is only used for the parasitic mass comparison here. The purpose of the presented method is to allow higher fidelity simulation of the solid materials, and it would not make sense to use a simplified TLM model in this case. This simplified model should not be confused with the co-simulated FE piston model analysed later in this paper.

The Hopsan simulation tool is a dedicated 1D TLM simulation software used primarily for simulation of fluid power and mechatronic systems. Physical systems are modelled using the TLM method and power ports are used as the interface between sub-models. A power port defines the variables that are necessary to describe power transfer in a specific domain. In Hopsan, they also contain other variables particular to that domain and the TLM method. Examples of available power ports for 1D power transfer are: hydraulic, translational mechanic, rotational mechanic, electric and pneumatic types. The use of power ports makes it possible to define physically motivated connection points similar to the ones that exist in the real world. For example, a hydraulic cylinder has hydraulic connections for the fluid pressure supply and a mechanical attachment on the piston that transfers the force to some load as shown in Fig. 5. Aside from power-port modelling, the program also supports scalar signal flow for arbitrary mathematics. Port types are then of input or output type. For the co-simulation method described, only the hydraulic and translational mechanic port types are considered.

In Hopsan, the TLM elements are referred to as C-type sub-models, since they calculate the characteristic waves. The boundary values of the element are solved by the so-called Q-type sub-models that calculate the flow variable Q. These two types are responsible for solving one part each of the TLM equation as illustrated by Eq. 9, and must therefore be connected pairwise as Fig. 5 shows. This grouping approach is the same as was recommended in [27]. The C-type models will introduce the approximations discussed previously, and must be implemented so that they calculate the characteristic waves. The Q-type models can be more freely implemented, and these usually contain their own internal integrators. For instance, the piston in Fig. 5 is essentially a mass with two surfaces subjected to fluid pressure given by c and \(Z_{c}\). It internally integrates the resulting accelerations into velocity (the flow variable Q) and position. Essentially, this is the same thing that happens in LS-DYNA, where the FE-model of the piston is simulated, and it is therefore suitable to let the FMU co-simulation module be of Q-type.

The general non-linear ODE, i.e. the equation of motion, that is solved by LS-DYNA is given by Eq. 10.where \(\mathbf M\) is the mass matrix, \(\mathbf C\) is the damping matrix and \(\mathbf f _\mathrm{Int}\) is the internal force vector. The external force vector, \(\mathbf f _\mathrm{Ext}\), represents all external loads acting on the FE-model. The displacement \(\mathbf u\) is solved explicitly by the central difference time integration scheme described by Eqs. 11 and 12. For further description regarding the explicit integration scheme, see e.g. [30]. The UDF functionality for adding external loads \(\mathbf f _\mathrm{Ext}\), is used as the interface on the LS-DYNA side. The input variables (c and \(Z_{c}\)), representing the pressure, from the system model are used to calculate these forces and the user-defined configuration file specifies on which locations they are to be applied. The resulting displacements and velocities are sent back from the FE-model to the system model. This procedure is repeated each time step until the analysis is completed.

To avoid needless complication of the FE-model, it is represented by a sub-model of Q-type in Hopsan, essentially as a replacement to the piston model in Fig. 5. This leaves the TLM specific modelling of the characteristic impedance and wave variables in Hopsan. The Q-type FE-model will determine the flow variables, in this case the velocity and the position (by integration), based on the input from the neighbouring C-type components in the system model. This results in a natural interface point between the models. In the application example, the hydraulic cylinder presented in Sect. 4, the hydraulic oil cavities (fluid-domain) or mechanic external forces are implemented as TLM elements in Hopsan. The fluid pressure or the external forces are then applied to the structure-domain piston surface segments and contribute to the external forces vector \(\mathbf f _\mathrm{Ext}\) in Eq. 10.

Since Hopsan is administering the simulation, it will trigger each simulation step in LS-DYNA. Hopsan uses fixed time step simulation and the FE-simulation is set to use the same step size. Technically, it would be possible to sub step the simulation in LS-DYNA, but that is deemed to be of little value as the time spent on the 1D system simulation is negligible compared to the FE-simulation time. The co-simulation sequence and synchronisation are shown in Fig. 6, where the UDF module in LS-DYNA is called at the beginning of each FE time step to set the external forces. The two programs exchange variables at this point. The LS-DYNA simulation loop always waits for input at the beginning of each step. The socket receive calls are blocking (with a failure timeout) and this effectively synchronises the simulation steps between the programs. Aside from communicating the simulation variables, control flags are also transmitted. This allows one program to notify the other when the simulation has been finished or otherwise terminated.

This case essentially integrates the effect of the hydraulic forces acting on the piston mass, i.e. the mechanical force is set to zero. The time step \(T=10^{-6}\) s was used here based on a convergence study of the Hopsan reference model. In this study, T was decreased until the amplitude error \(\epsilon\) was within acceptable levels. \(\epsilon\) was defined according to Eq. 13, where \(u_{T_{1}}\) is the piston displacement from the simulation at time step \(T_{1}\). RMS represents the Root-Mean-Square value. Piston displacements were extracted from simulations made with two different time step values, e.g. \(T_{1}=10^{-4}\) s and \(T_{2}=10^{-5}~s\). The time step was decreased one decade at a time and an acceptable level was reached for \(T_{1}=10^{-6}\) s where \(\epsilon <0.02~\%\).The volume was \(8.63 \times 10^{-6}\) m\(^{3}\) and \(2.05 \times 10^{-6}\) m\(^{3}\), for Cavity A and B, respectively. The characteristic impedances \(Z_{c,\rm fluid}\) were retrieved for each cavity from Hopsan; \(3.09 \times 10^{8}\) Pa s/m\(^{3}\) and \(8.68 \times 10^{8}\) Pa s/m\(^{3}\). The parasitic masses added due to parasitic inductance were estimated to be \(1.4 \times 10^{-4}\) kg and \(3.3 \times 10^{-5}\) kg. This amount of parasitic mass can be neglected compared to the actual piston mass in the model (11.2 kg). The computational time for the co-simulation was approximately 8 min and for the pure Hopsan reference model 0.3 s. Simulation results are shown in Fig. 9.

Figure 9a, b shows that the piston moves in and out of the cylinder housing at a maximum speed of \(\sim\)5 m/s. The results shows an identical behaviour for the reference and the co-simulation model. This was to be expected since the two-system models are the same except for the piston movement simulation. In Fig. 9c, it can be seen that the curves are right on top of each other, but when the curves are examined at a greater magnification, a maximum time shift of \(\sim\)0.13 ms can be noticed at the last pressure peak, see Fig. 10. It was also found that the magnitude of the time shift is increasing throughout the simulation.

The pressure curve, Fig. 9c, shows that the simulation method is able to handle short duration dynamics in the fluid system, causing pressure peaks of high amplitude and a duration of approximately 0.5 ms. The results also show that the piston reaches a steady-state behaviour after two working cycles, which implies that this method is stable for this simulation model. The FE-results in this case consist of the piston movement and the forces that are acting on the piston.

The second case is used to evaluate the simulation method when there is a transient mechanical force acting on the FE-model. The force is here used to simulate a simplified elastic impact, and will affect the pressure and flow in the hydraulic circuit. The magnitude of the force was chosen arbitrarily to \(10^{7}\) N to give a reasonable behaviour of the system, and its value thus has no direct physical basis. The time step used was \(10^{-7}\) s resulting in an estimated amplitude error of <7.5 % based on a convergence study in the same way as in Case 1. A significant part of this error can be related to the modelling of the mechanical force, where the time step has a large influence on the duration of the force.

The volume of the fluid cavities has here been changed to achieve a more realistic behaviour of the hydraulic percussion unit. The volumes that were used for Cavity A and B were: \(8.63 \times 10^{-4}\) m\(^{3}\) and \(2.05 \times 10^{-4}\) m\(^{3}\), respectively. As in Case 1, the characteristic impedances were retrieved from Hopsan; \(3.09 \times 10^{5}\) Pa\(\times\)s/m\(^{3}\) and \(8.68 \times 10^{5}\) Pa s/m\(^{3}\). The parasitic masses were estimated to be \(1.4 \times 10^{-8}\) kg and \(3.3 \times 10^{-9}\) kg, which are even lower than in Case 1 and can be considered negligible. The computational time for the co-simulation was approximately 29 min and for the pure Hopsan reference model 0.8 s; the simulation results can be seen in Fig. 11.

The piston displacement curve, Fig. 11a, shows the piston working sequence. At a position of 0.03 m, the piston reaches the inner turning point in the cylinder housing and then it starts to accelerate outwards in the cylinder due to the pressure in Cavity A. When the piston reaches a position of 0.085 m, the mechanical force is activated. The force will first quickly stop the piston and then accelerate it back into the cylinder housing. When the piston position is below 0.085 m, the force is deactivated and the forces from the hydraulic pressure will control the piston movement entirely. The duration of the applied mechanical force was found to be \(\sim\)22 µs. The piston velocity curve, Fig. 11b, shows that the piston is accelerated to a velocity of \(\sim\)9.5 m/s before the mechanical force is activated. Figure 11c shows that the pressure in Cavity A is \(\sim\)15 MPa when the piston is at the inner turning point. During the acceleration phase, the pressure drops to \(\sim\)13 MPa before the mechanical force is activated. At region M, the pressure is transient and it peaks up to 19 MPa. This is due to the activation of the mechanical force and the response from the fluid system. The maximum value occurs when the piston moves into the cylinder housing and the valve is closed when changing state. In Fig. 11d, the momentary flow into Cavity A can be seen. During the acceleration phase, the flow increases to 0.008 m\(^{3}\)/s before the mechanical force is activated, after that the flow is reversed and is decreased towards a zero flow at the inner turning point.

The magnifier in Fig. 11c shows that the curves are close to each other; a maximum time shift difference of 0.9 ms was noticed during the whole simulation time period. Note that the time shift varies throughout the analysis. The maximum pressure magnitude difference was estimated to 0.5 %, where the co-simulation pressure in general is somewhat lower compared to the reference model.

The third simulation case was chosen to demonstrate the potential of the proposed method and the possibilities when evaluating deformations, stresses and strains in the structural parts. The model in Fig. 7b was used for the system simulation. The previous FE-model was updated to include a cylindrical part that represents a typical tool for a hydraulic hammer, see Fig. 12. This set-up is used to give a more realistic impact response than the mechanical force used in Case 2. The far end from the impact surface of the tool was rigidly fixed. The elastic material properties stated in Table 2 were used in the FE-model to simulate the elastic behaviour. The mesh for the tool consists of 5114 fully integrated 8-node hexahedral solid elements. The total number of elements was 12,256. The time step used here was \(10^{-7}\) s, as in Case 2, resulting in a computational time of 4 hours and 44 minutes. The other parameters in the Hopsan model were also the same as in Case 2.

The results, shown in Fig. 13, indicate that the percussion unit reaches a steady-state condition after only a few working strokes. The values for the piston movement are calculated by averaging over the nodes on the piston impact surface. The piston displacement curve, Fig. 13a, shows that the piston follows the same sequence as described in Case 2. The difference is that a real impact is simulated in this case, instead of applying a mechanical force. The impact on the tool occurs at a position of 0.085 m. A stress wave is generated at the piston impact surface at the time of impact, and the noise in the velocity curve, see Fig. 13b, after the first impact, represents the stress wave travelling back and forth in the piston. The pressure curve, Fig. 13c, is very similar to the curve in Case 2 and it follows the same pattern. The hydraulic pressure in Cavity A shows a rapid variation at the time of impact, see region N in Fig. 13c. At this time, the piston “bounces” on the tool and the piston moves away from the tool. This movement will cause a pressure increase in the fluid contained in Cavity A, since the piston is moving back into the cavity in a similar way as in Case 2. As the pressure increases in the cavity, the fluid force on the piston will also increase, and this will affect the piston movement.

Figures 14 and 15 show some examples of FE-results that are available from the simulation. In this case, the equivalent von Mises stress is shown, but also displacements are available. The transient stress curve in Fig. 15 exhibits the same behaviour as the pressure curve from Cavity A, cf. Fig. 13c.