Mass-spring system (computer graphics)

The mass-spring system (short MFS , English mass-spring-system ), in the literature also mass-spring model (short MFM ), is a computer graphics method to determine the deformation behavior of objects based on polygon meshes . It can be applied to both flat and spatial objects.

The reasons for the widespread use of the mass-spring system are the easily understandable physical concept, the simple implementation and the relatively low computational requirements.

history

The mass-spring system is based on the physical approaches of Hooke's law .

Because of the low computing power of computers at the time, it was initially only used for surfaces (both two-dimensional and three-dimensional surfaces, for example a tablecloth that falls over a table). Therefore, when it was first used in 1981, it was initially only used to simulate skin and fabric behavior. It was not until 1989 that the first MFS solid models were published.

Today it is a common method in the computer game industry and in medicine for simulating operative interventions.

construction

Two ground nodes connected by a spring and a damper.

In a mass-spring system, objects are represented by a network of mass nodes, mechanical springs and dampers.

The process of moving a point in a mass-spring system at different points in time.

The nodes of a polygon network are each given a mass . The edges are represented by a mechanical spring , which is connected in parallel with a damping element. Thus it has a spring constant , a rest length and a decay constant . The attenuator is necessary if the object does not vibrate after deformation, but should find a position of rest. The coefficient of friction is calculated depending on the mass nodes .

The ground nodes are regularly distributed throughout the object and represent it that way. Springs are placed between them to connect the ground nodes. If two ground nodes are connected to one another, they are called adjacent. The initial shape of the object is when all springs are in the rest position ( ), i.e. have their rest length. If a ground node is now moved, a potential energy is applied to the neighboring springs ( ), which is now minimized by moving the neighboring mass points. Such a shift affects the entire network, causing it to shift. ${\displaystyle E_{\mathrm {pot} }=0}$ ${\displaystyle E_{\mathrm {pot} }\neq 0}$

In order to counteract a displacement in space, points can be "fixed" (at an absolute position or at a position relative to another mass point). In this way, among other things, plastic deformation can be achieved.

To make physical behavior more realistic, the mass-spring system can be supplemented by collision detection so that the mass points can react to collisions with one another and with other objects.

calculation

The following derivation is based on a -dimensional space. ${\displaystyle d}$

Mass-spring systems have their origin in classical mechanics . They are based on the idea that a body can be represented by a single ground node . Furthermore, this ground node can be divided into smaller ground nodes , distributed over the whole body, so that: ${\displaystyle M}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m_{i}}$

${\displaystyle m=\sum \limits _{i=1}^{n}m_{i}}$

According to Hooke's law, the behavior of a spring with a parallel attenuator is defined as follows:

${\displaystyle {\vec {F}}_{s}=-k\cdot {\vec {s}}-\mu \cdot {\dot {\vec {s}}}}$

where is the spring constant, the deflection of the spring to the rest position and the coefficient of friction. ${\displaystyle k}$ ${\displaystyle {\vec {s}}}$ ${\displaystyle \mu }$

Newton's definition of the behavior of a moving mass is:

${\displaystyle {\vec {F}}=m\cdot {\vec {a}}}$

${\displaystyle {\vec {F}}}$is the force that gives the body of the mass the acceleration . ${\displaystyle m}$ ${\displaystyle {\vec {a}}}$

On the basis of these last two equations, the equation of motion of an individual mass node can be derived. ${\displaystyle i}$

The force is exerted on the mass node with the mass during the displacement : ${\displaystyle i}$ ${\displaystyle m_{i}}$ ${\displaystyle {\vec {s}}_{i}\in \mathbb {R} ^{d}}$ ${\displaystyle {\vec {F}}_{i}}$

${\displaystyle {\vec {F}}_{i}=m_{i}\cdot {\ddot {{\vec {s}}_{i}}}}$

The deflection is calculated with two mass nodes and in a mass-spring system with the distance from each other, minus the length of the spring at rest : ${\displaystyle {\vec {s}}_{ij}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle l_{0,ij}}$

${\displaystyle {\vec {s}}_{ij}={\frac {{\vec {x}}_{j}-{\vec {x}}_{i}}{\begin{vmatrix}{\vec {x}}_{j}-{\vec {x}}_{i}\end{vmatrix}}}\cdot ({\begin{vmatrix}{\vec {x}}_{j}-{\vec {x}}_{i}\end{vmatrix}}-l_{0,ij})}$

${\displaystyle {\vec {x}}_{i}}$is the position of the -th ground node. The direction vector is calculated as a unit vector from ground node to ground node . The amount of is the distance between the ground node and the ground node minus the rest length of the spring, which is the length scalar. The scalar product is the displacement of the ground node as a function of the ground node . ${\displaystyle i}$ ${\displaystyle {\frac {{\vec {x}}_{j}-{\vec {x}}_{i}}{\begin{vmatrix}{\vec {x}}_{j}-{\vec {x}}_{i}\end{vmatrix}}}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle {\begin{vmatrix}{\vec {x}}_{j}-{\vec {x}}_{i}\end{vmatrix}}-l_{0,ij}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle {\vec {s}}_{ij}}$ ${\displaystyle i}$ ${\displaystyle j}$

This leads to the following force, which acts on the individual ground node with the neighboring node : ${\displaystyle i}$ ${\displaystyle j}$

${\displaystyle {\vec {F}}_{s,ij}=-k_{ij}\cdot {\vec {s}}_{ij}-\mu _{ij}{\dot {{\vec {s}}_{i}}}}$

The external forces acting on the ground node can be summarized as: ${\displaystyle i}$ ${\displaystyle {\vec {F}}_{\mathrm {ext} ,i}}$

${\displaystyle {\vec {F}}_{\mathrm {ext} ,i}={\vec {F}}_{G,i}+{\vec {F}}_{e,i}}$

with the vector of the external force , the gravitational vector and the vector of the other external influences . ${\displaystyle {\vec {F}}_{\mathrm {ext} ,i}}$ ${\displaystyle {\vec {F}}_{G,i}}$ ${\displaystyle {\vec {F}}_{e,i}}$

If you summarize all forces acting on the mass node , you get the following differential equation as an equation of motion, where the set of neighboring mass nodes is: ${\displaystyle i}$ ${\displaystyle N}$

${\displaystyle {\vec {F}}_{i}=\sum \limits _{j\in N}{\vec {F}}_{s,ij}+{\vec {F}}_{\mathrm {ext} ,i}}$

The following applies : ${\displaystyle {\vec {F}}_{i}=m_{i}{\ddot {{\vec {s}}_{i}}}}$

${\displaystyle m_{i}{\ddot {{\vec {s}}_{i}}}=\sum \limits _{j\in N}{\vec {F}}_{s,ij}+{\vec {F}}_{\mathrm {ext} ,i}}$

The equation of motion for the entire system is obtained by describing all displacements with the vector , the masses with the matrix , the coefficients of friction with the matrix and the spring constants of the individual mass nodes through the matrix : ${\displaystyle s_{i}}$ ${\displaystyle {\vec {s}}=({\vec {s}}_{1},{\vec {s}}_{2},\dots ,{\vec {s}}_{n})^{T}}$ ${\displaystyle m_{i}}$ ${\displaystyle M}$ ${\displaystyle \mu _{ij}}$ ${\displaystyle D}$ ${\displaystyle k_{ij}}$ ${\displaystyle K}$

${\displaystyle {\vec {F}}_{\mathrm {ext} }=M\cdot {\ddot {\vec {s}}}+D\cdot {\dot {\vec {s}}}+K\cdot {\vec {s}}}$

The matrices and are both symmetrical, da and . It follows that not all components but only components have to be calculated. In addition, the components on the main diagonal of the matrices and (i.e. the components and ) can be neglected because there is no spring connected to the same ground node at both ends. The number of components to be calculated is thus reduced to . ${\displaystyle D}$ ${\displaystyle K}$ ${\displaystyle \mu _{ij}=\mu _{ji}}$ ${\displaystyle k_{ij}=k_{ji}}$ ${\displaystyle n^{2}}$ ${\displaystyle {\frac {n^{2}+3n}{2}}}$ ${\displaystyle D}$ ${\displaystyle K}$ ${\displaystyle \mu _{ii}}$ ${\displaystyle k_{ii}}$ ${\displaystyle {\frac {n^{2}+n}{2}}}$

See also

• Spring pendulum
• Harmonic oscillator

Web links

Commons : mass-spring systems  - collection of images, videos and audio files

• Johan Jansson: Examples of the mass-spring system

Individual evidence

1. ↑ ^{a b c} Markus A. Schill: Biomechanical Soft Tissue Modeling Techniques, Implementation and Applications . University of Mannheim, Mannheim 2001 (English, PDF, 24.6MB [accessed on July 1, 2011] dissertation).
2. ^ ^{A b} S. Platt, N. Badler: Animating Facial Expressions . In: Computer Graphics . tape 15 , no. 3 , 1981 (English, PDF ).
3. ↑ K. Waters: A Muscle Model for Animating Three-Dimensional Facial Expression . In: Computer Graphics . tape 21 , no. 4 , 1987 (English, Link ).
4. ^ J. Chadwick, D. Haumann, R. Parent: Layered Construction for Deformable Animated Characters . In: Proceedings of ACM SIGGRAPH . 1989 (English, PDF ).
5. ↑ D. Terzopoulos, K. Waters: Physically-based Facial Modeling, Analysis and Animation . In: The Journal of Visualization and Computer Animation . tape 1 , 1990 (English, PDF ). PDF ( Memento of the original from April 23, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
6. ↑ ^{a b} Jens Neumann: Process for adhoc modeling and simulation of spatial spring-mass systems for use in virtual reality-based handling simulations . Technische Universität Berlin, Fraunhofer IRB Verlag, 2009, ISBN 978-3-8167-7954-4 (dissertation).