**2. Basics of model building and simulation**

Computer modeling is a widely used method for determining the biomechanical behavior of a system. A distinction is made between the FE modeling and MBS modeling. Both simulation methods are used to aid engineers and researchers in investigating and analyzing mechanical and mechatronic systems in various fields such as the car, railway, engine, aerospace indus‐ tries, and medical applications. The difference between the FE modeling and MBS modeling lies in the basic model structure, and thus in the field of application. For analyzing highly sophisticated problems, the FE modeling is the appropriate modeling method. Within this mathematical method, based on the numerical solution of partial differential equations, the system is divided into a finite number of simple geometric elements, the finite elements [12]. An FE model is composed of geometric elements starting with points in space, which are linked to each other by so-called edges. Three points define a triangular plane. The corners are called vertices. Connected triangles result in faces which have a contour defined by a sequence of vertices, which is called a polygon. For visualization purposes, the faces can be colored and

**Figure 1.** Different elements of polygon mesh modeling.

textured. This information is attached to the vertices, as it is common in computer graphics (**Figure 1**).

a conservative treatment does not achieve a pain relief anymore, only a surgically induced stabilizationlike a therapeuticoptionremains.Mayer[6]indicates thatnovalidatedresults exist aboutthe successoftheseoperationsover aperiodoftwoyears, andthusprovide clear evidence ofreduced adjacent segment degeneration. Furthermore, statements aboutthe sustainability of clinical and radiological results with respect to a reconstruction of the segmental mobility are still missing. A modern non-invasive method for assessing the effect of medical operational measures is theloaddeterminationinvarious spinal structuresbymeansof computermodeling.

To capture the load distribution of the lumbar spine, highly developed so-called finite-element (FE) models [7–11] can be used which allow a very detailed calculation of force and torque transmission—both in the rigid bones and in the elastic proportions as intervertebral discs, ligaments, and implants. The complexity of the models, however, requires high computing times for each load case. Is the aspect of the holistic approach of the human movement in the focus of interest, the multibody simulation (MBS) is a suitable simulation method? Due to the highly efficient short computation times, even complex motion sequences can be simulated.

Computer modeling is a widely used method for determining the biomechanical behavior of a system. A distinction is made between the FE modeling and MBS modeling. Both simulation methods are used to aid engineers and researchers in investigating and analyzing mechanical and mechatronic systems in various fields such as the car, railway, engine, aerospace indus‐ tries, and medical applications. The difference between the FE modeling and MBS modeling lies in the basic model structure, and thus in the field of application. For analyzing highly sophisticated problems, the FE modeling is the appropriate modeling method. Within this mathematical method, based on the numerical solution of partial differential equations, the system is divided into a finite number of simple geometric elements, the finite elements [12]. An FE model is composed of geometric elements starting with points in space, which are linked to each other by so-called edges. Three points define a triangular plane. The corners are called vertices. Connected triangles result in faces which have a contour defined by a sequence of vertices, which is called a polygon. For visualization purposes, the faces can be colored and

**2. Basics of model building and simulation**

30 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 1.** Different elements of polygon mesh modeling.

For each junction of elements, the so-called nodes, boundary, and transition conditions are defined in accordance with a specific material law [13]. In this way, highly detailed analyses of material properties can be carried out by an FE model. However, FE modeling requires a relatively long computational time. According to Berkley [14], the model's accuracy increases with the number of finite elements that are used to describe the geometry. But each additional element also means additional computational time. Solving such large numbers of FE equa‐ tions leads to an enormously time-consuming calculation as in [15, 16]. The accuracy of the system and the expected computing time must therefore be carefully matched. A much faster calculation method is the MBS. The MBS describes the interaction of individual bodies with each other and with their environment, that is, the forces and torques acting on the bodies and between the bodies as well as the resulting movements [17].

The modeled bodies are assumed to be rigid and thus not deformable. The different bodies are linked through massless joints and kinematic constraints, which allow certain relative motions and restrict others [12]. Each body has a specific mass and a moment of inertia. This mass and inertia characteristics are associated with a single point, the center of gravity of the body. Thereby, every rigid body is provided with its own coordinate system (reference system). Based on this coordinate system, the exact location in space can be determined with respect to the inertial frame.

The degrees of freedom (DoF) of a rigid body are described by independent coordinates. A rigid body has a maximum of six DoF. It can move along the *x*-, *y*-, and *z*-axis (**Figure 2**, left), and rotate about these axes (**Figure 2**, right).

**Figure 2.** Degrees of freedom of a rigid body [18]: A rigid body is free to move along the *x*-, *y*-, and *z*-axis (left) and rotate about the *x*-, *y*-, and *z*-axis (right).

Possible translations and rotations relative to the inertial system can be realized through the implementation of appropriate joint types. If a model is composed of two or more rigid bodies, they are connected by a massless joint. The joints provide certain DoF. Depending on the definition of joint type, the joint


Furthermore, it is possible that forces and torques can be transferred between two body markers, which are generally located on different bodies, by a so-called force element. The type of interaction depends on the implemented force law. According to [19], the DoF of a model are determined by a number of independent state variables, which characterize the movement of the body. Furthermore, the movement of the MBS system depends on the inertial properties of its bodies, masses, inertia tensors, and its centers of gravity. The kinematics of the model is defined by a system of coupled differential equations. These equations of motion give the relation between motion and forces. For this reason, they can be solved in two ways: the forward or direct dynamics and the inverse dynamics. In the case of forward dynamics, the movement is determined by known internal forces or torques, whereas the reconstruction of the internal forces or torques from movements and external forces, like the gravity, is called inverse dynamics [20].

It should be noted that the presented modeling styles MBS and FE can be used independently of each other, but can also be combined with each other.
