Crash simulation
Encyclopedia
A crash simulation is a virtual
recreation of a destructive crash test
of a car
or a highway guard rail system
using a computer simulation
in order to examine the level of safety of the car and its occupants. Crash simulation
s are used by automaker
s during Computer-Aided Engineering
(CAE) analysis for crashworthiness
in the Computer-Aided Design
(CAD) process of modelling new cars. During a crash simulation, the kinetic energy
, energy of motion, that a vehicle
has before the impact is transformed into deformation energy
, mostly by plastic deformation (plasticity
) of the car body material
(Body in White
), at the end of the impact.
Data obtained from a crash simulation indicate the capability of the car body or guard rail structure to protect the vehicle occupants during a collision
(and also pedestrian
s hit by a car) against injury
. Important results are the deformations (for example, steering wheel
intrusions) of the occupant space (driver, passenger
s) and the decelerations (for example, head acceleration) felt by them, which must fall below threshold values fixed in legal car safety
regulations. To model real crash tests, today's crash simulations include virtual models of crash test dummies
and of passive safety devices (seat belts, airbags, shock absorbing dash boards, etc.). Guide rail tests evaluate vehicle deceleration and rollover potential, as well as penetration of the barrier by vehicles.
, which require as an input the results of physical destructive laboratory tests, needed to determine the mechanical crushing behavior of each spring component of the modeled system. "First principle" simulations like more elaborate finite element models, however, need only the definition of the structural geometry and the basic material properties (rheology
of car body steel, glass, plastic parts, etc.) as an input to generate the numerical model.
The origins of industrial first principle computerized car crash simulation lie in military defense, outer space
and civil nuclear power plant
applications. Upon presentation of a simulation of the accidental crash of a military fighter
plane into a nuclear power plant on May 30, 1978 by ESI Group in a meeting organized by the Verein Deutscher Ingenieure
(VDI) in Stuttgart, car makers became alerted to the possibility of using this technology for the simulation of destructive car crash tests (Haug 1981).
In the following years, German car makers produced more complex crash simulation studies, simulating the crash behavior of individual car body components, component assemblies, and quarter and half car bodies in white (BIW
). These experiments culminated in a joint project by the Forschungsgemeinschaft Automobil-Technik (FAT), a conglomeration of all seven German car makers (Audi
, BMW
, Ford
, Mercedes-Benz
, Opel
, Porsche
, and Volkswagen
), which tested the applicability of two emerging commercial crash simulation codes. These simulation codes recreated a frontal impact of a full passenger car structure (Haug 1986) and they ran to completion on a computer overnight. Now that turn-around time between two consecutive job-submissions (computer runs) did not exceed one day, engineers were able to make efficient and progressive improvements of the crash behavior of the analyzed car body structure.
of the car occupants during impacts on the front end structure of the car in a "head-on collision
" or "frontal impact", the lateral structure of the car in a “side collision
” or “side impact”, the rear end structure of a car in a "rear-end collision
" or “rear impact”, and the roof structure of the car when it overturns during a "rollover
". Crash simulations can also be used to assess injury to pedestrians hit by a car.
of a new car model. This way, tests can be performed quickly and inexpensively in a computer, which permits optimization of the design before a real prototype of the car has been manufactured. Using a simulation, problems can be solved before spending time and money on an actual crash test. The great flexibility of printed
output and graphical display enables designers to solve some problems that would have been nearly impossible without the help of a computer.
Crash simulations use a method of analysis called the finite element method
to solve complex problems by dividing a surface into a large but still finite number of elements and determining the motion of these elements over very small periods of time. Essentially, using this method, one can solve many smaller problems instead of one larger one.
PAM-CRASH
started crash simulation and together with LS-DYNA
is a software package which is widely used for crash simulation.
, that is, breaking up the continuous movement of the body in real time into smaller changes in position over small, discrete time steps. The discretization involves subdividing the surface of the constituent, thin, sheet metal
parts into a large number (approaching one million in 2006) of quadrilateral
or triangular regions, each of which spans the area between "nodes" to which its corners are fixed. Each element has mass, which is distributed as concentrated masses and as mass moments of inertia
to its connecting nodes. Each node has 6 kinematic degrees of freedom
, that is, one node can move in three linear directions under translation
and can rotate
about three independent axes. The spatial coordinates (x), displacement (u), velocity
(v), and acceleration
(a) of each node is mostly expressed in a three-dimensional rectangular Cartesian coordinate system
with axes X,Y, and Z.
If the nodes move during a crash simulation, the connected elements move, stretch, and bend with their nodes, which causes them to impart forces and moments
to their nodal connections. The forces and moments at the nodes correspond to the inertia forces and moments, caused by their translational (linear) and angular acceleration
s and to the forces and moments transmitted by the resistance
of the structural material of the connected elements as they deform. Sometimes, additional external structural loads are applied, like gravity loads from the self weight of the parts, or added loads from external masses.
The forces and moments of all nodes are collected into a column vector (or column matrix), and the time dependent equations of motion
(in dynamic equilibrium) can be written as follows.
where vector
(mass times acceleration vector) collects the inertia forces at the nodes, 
collects the external nodal loads, and 
collects the internal resisting forces from the deformation of the material. M is a diagonal matrix
of the nodal masses. Each vector (u, v, a, F, etc.) has dimension
6 times the total number of nodes in the crash model (about 6 million “degrees of freedom
” for every 1 million "nodes" in 3-D thin shell finite element models).
time integration method used by most crash codes, the accelerations, velocities, and displacements of the body are related by the following equations.
In these equations the subscripts n±1/2, n, n+1 denote past, present, and future times, t, at half and full time intervals with time steps
and 
, respectively.
, the velocities, 
, and the displacements, 
, at each discrete point in time, t, during the crash duration. This solution is trivial, since the mass matrix is diagonal. The computer time is proportional to the number of finite elements and the number of solution time steps. The stable solution time step, 
, is limited for numerical stability
, as expressed by the Courant Fredericks Levy condition(CFL), which states that “in any time-marching computer simulation, the time step must be less than the time for some significant action to occur, and preferably considerably less." In a crash simulation, the fastest significant actions are the acoustic signals that travel inside the structural material.
The solid elastic stress wave speed amounts to
where
is the initial elastic modulus (before plastic deformation) of the material and 
is the mass density. The largest stable time step for a given material is therefore
,
where
is the smallest distance between any two nodes of the numerical crash simulation model.
Since this distance can change during a simulation, the stable time step changes and must be updated continually as the solution proceeds in time. When using steel
, the typical value of the stable time step is about one microsecond
when the smallest discrete node distance in the mesh of the finite element model is about 5 millimeters. It needs then more than 100,000 time intervals to solve a crash event that lasts for one tenth of a second. This figure is exceeded in many industrial crash models demanding optimized crash solvers with High-Performance Computing (HPC
) features, such as vectorization
and parallel computing
.
Virtual
The term virtual is a concept applied in many fields with somewhat differing connotations, and also, differing denotations.The term has been defined in philosophy as "that which is not real" but may display the salient qualities of the real....
recreation of a destructive crash test
Crash test
A crash test is a form of destructive testing usually performed in order to ensure safe design standards in crashworthiness and crash compatibility for various modes of transportation or related systems and components.- Types :...
of a car
Čar
Čar is a village in the municipality of Bujanovac, Serbia. According to the 2002 census, the town has a population of 296 people.-References:...
or a highway guard rail system
Traffic barrier
Traffic barriers keep vehicles within their roadway and prevent vehicles from colliding with dangerous obstacles. Traffic barriers installed at the road side also prevent errant vehicles from traversing steep slopes. Traffic barriers installed at the medians of divided highways are also referred...
using a computer simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...
in order to examine the level of safety of the car and its occupants. Crash simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
s are used by automaker
Automaker
The automotive industry designs, develops, manufactures, markets, and sells motor vehicles, and is one of the world's most important economic sectors by revenue....
s during Computer-Aided Engineering
Computer-aided engineering
Computer-aided engineering is the broad usage of computer software to aid in engineering tasks. It includes computer-aided design , computer-aided analysis , computer-integrated manufacturing , computer-aided manufacturing , material requirements planning , and computer-aided planning .- Overview...
(CAE) analysis for crashworthiness
Crashworthiness
Crashworthiness is the ability of a structure to protect its occupants during an impact. This is commonly tested when investigating the safety of aircraft and vehicles. Depending on the nature of the impact and the vehicle involved, different criteria are used to determine the crashworthiness of...
in the Computer-Aided Design
Computer-aided design
Computer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...
(CAD) process of modelling new cars. During a crash simulation, the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
, energy of motion, that a vehicle
Vehicle
A vehicle is a device that is designed or used to transport people or cargo. Most often vehicles are manufactured, such as bicycles, cars, motorcycles, trains, ships, boats, and aircraft....
has before the impact is transformed into deformation energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
, mostly by plastic deformation (plasticity
Plasticity (physics)
In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...
) of the car body material
Material
Material is anything made of matter, constituted of one or more substances. Wood, cement, hydrogen, air and water are all examples of materials. Sometimes the term "material" is used more narrowly to refer to substances or components with certain physical properties that are used as inputs to...
(Body in White
Body in White
Body in white or BIW refers to the stage in automotive design or automobile manufacturing in which a car body's sheet metal components have been welded together — but before moving parts the motor, chassis sub-assemblies, or trim have been added and before painting.The name...
), at the end of the impact.
Data obtained from a crash simulation indicate the capability of the car body or guard rail structure to protect the vehicle occupants during a collision
Collision
A collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...
(and also pedestrian
Pedestrian
A pedestrian is a person traveling on foot, whether walking or running. In some communities, those traveling using roller skates or skateboards are also considered to be pedestrians. In modern times, the term mostly refers to someone walking on a road or footpath, but this was not the case...
s hit by a car) against injury
Injury
-By cause:*Traumatic injury, a body wound or shock produced by sudden physical injury, as from violence or accident*Other injuries from external physical causes, such as radiation injury, burn injury or frostbite*Injury from infection...
. Important results are the deformations (for example, steering wheel
Steering wheel
A steering wheel is a type of steering control in vehicles and vessels ....
intrusions) of the occupant space (driver, passenger
Passenger
A passenger is a term broadly used to describe any person who travels in a vehicle, but bears little or no responsibility for the tasks required for that vehicle to arrive at its destination....
s) and the decelerations (for example, head acceleration) felt by them, which must fall below threshold values fixed in legal car safety
Car safety
Automobile safety is the study and practice of vehicle design, construction, and equipment to minimize the occurrence and consequences of automobile accidents. Automobile safety is the study and practice of vehicle design, construction, and equipment to minimize the occurrence and consequences of...
regulations. To model real crash tests, today's crash simulations include virtual models of crash test dummies
Crash Test Dummies
The Crash Test Dummies is a Canadian folk rock/alternative rock band from Winnipeg, Manitoba, widely known for their 1993 single "Mmm Mmm Mmm Mmm".The band is most identifiable through Brad Roberts and his distinctive bass-baritone voice...
and of passive safety devices (seat belts, airbags, shock absorbing dash boards, etc.). Guide rail tests evaluate vehicle deceleration and rollover potential, as well as penetration of the barrier by vehicles.
Origin
In the years 1970 attempts were made to simulate car crash events with non-linear spring-mass systems after calibrationCalibration
Calibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....
, which require as an input the results of physical destructive laboratory tests, needed to determine the mechanical crushing behavior of each spring component of the modeled system. "First principle" simulations like more elaborate finite element models, however, need only the definition of the structural geometry and the basic material properties (rheology
Rheology
Rheology is the study of the flow of matter, primarily in the liquid state, but also as 'soft solids' or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force....
of car body steel, glass, plastic parts, etc.) as an input to generate the numerical model.
The origins of industrial first principle computerized car crash simulation lie in military defense, outer space
Outer space
Outer space is the void that exists between celestial bodies, including the Earth. It is not completely empty, but consists of a hard vacuum containing a low density of particles: predominantly a plasma of hydrogen and helium, as well as electromagnetic radiation, magnetic fields, and neutrinos....
and civil nuclear power plant
Nuclear power plant
A nuclear power plant is a thermal power station in which the heat source is one or more nuclear reactors. As in a conventional thermal power station the heat is used to generate steam which drives a steam turbine connected to a generator which produces electricity.Nuclear power plants are usually...
applications. Upon presentation of a simulation of the accidental crash of a military fighter
Fighter aircraft
A fighter aircraft is a military aircraft designed primarily for air-to-air combat with other aircraft, as opposed to a bomber, which is designed primarily to attack ground targets...
plane into a nuclear power plant on May 30, 1978 by ESI Group in a meeting organized by the Verein Deutscher Ingenieure
Verein Deutscher Ingenieure
Verein Deutscher Ingenieure is an organization of 139,000 engineers and natural scientists.Established in 1856, the VDI is today the largest engineering association in Western Europe....
(VDI) in Stuttgart, car makers became alerted to the possibility of using this technology for the simulation of destructive car crash tests (Haug 1981).
Body in White
Body in white or BIW refers to the stage in automotive design or automobile manufacturing in which a car body's sheet metal components have been welded together — but before moving parts the motor, chassis sub-assemblies, or trim have been added and before painting.The name...
). These experiments culminated in a joint project by the Forschungsgemeinschaft Automobil-Technik (FAT), a conglomeration of all seven German car makers (Audi
Audi
Audi AG is a German automobile manufacturer, from supermini to crossover SUVs in various body styles and price ranges that are marketed under the Audi brand , positioned as the premium brand within the Volkswagen Group....
, BMW
BMW
Bayerische Motoren Werke AG is a German automobile, motorcycle and engine manufacturing company founded in 1916. It also owns and produces the Mini marque, and is the parent company of Rolls-Royce Motor Cars. BMW produces motorcycles under BMW Motorrad and Husqvarna brands...
, Ford
Ford Motor Company
Ford Motor Company is an American multinational automaker based in Dearborn, Michigan, a suburb of Detroit. The automaker was founded by Henry Ford and incorporated on June 16, 1903. In addition to the Ford and Lincoln brands, Ford also owns a small stake in Mazda in Japan and Aston Martin in the UK...
, Mercedes-Benz
Mercedes-Benz
Mercedes-Benz is a German manufacturer of automobiles, buses, coaches, and trucks. Mercedes-Benz is a division of its parent company, Daimler AG...
, Opel
Opel
Adam Opel AG, generally shortened to Opel, is a German automobile company founded by Adam Opel in 1862. Opel has been building automobiles since 1899, and became an Aktiengesellschaft in 1929...
, Porsche
Porsche
Porsche Automobil Holding SE, usually shortened to Porsche SE a Societas Europaea or European Public Company, is a German based holding company with investments in the automotive industry....
, and Volkswagen
Volkswagen
Volkswagen is a German automobile manufacturer and is the original and biggest-selling marque of the Volkswagen Group, which now also owns the Audi, Bentley, Bugatti, Lamborghini, SEAT, and Škoda marques and the truck manufacturer Scania.Volkswagen means "people's car" in German, where it is...
), which tested the applicability of two emerging commercial crash simulation codes. These simulation codes recreated a frontal impact of a full passenger car structure (Haug 1986) and they ran to completion on a computer overnight. Now that turn-around time between two consecutive job-submissions (computer runs) did not exceed one day, engineers were able to make efficient and progressive improvements of the crash behavior of the analyzed car body structure.
Application
Crash simulations are used to investigate the safetySafety
Safety is the state of being "safe" , the condition of being protected against physical, social, spiritual, financial, political, emotional, occupational, psychological, educational or other types or consequences of failure, damage, error, accidents, harm or any other event which could be...
of the car occupants during impacts on the front end structure of the car in a "head-on collision
Head-on collision
A head-on collision is one where the front ends of two ships, trains, planes or vehicles hit each other, as opposed to a side collision or rear-end collision.-Rail transport:...
" or "frontal impact", the lateral structure of the car in a “side collision
Side collision
Side collisions are vehicle crashes where the side of one or more vehicles is impacted. These crashes often occur at intersections, in parking lots, when two vehicles pass on a multi-lane roadway, or when a vehicle hits a fixed object....
” or “side impact”, the rear end structure of a car in a "rear-end collision
Rear-end collision
A rear-end collision is a traffic accident wherein a vehicle crashes into the vehicle in front of it, usually caused by tailgating or panic stops...
" or “rear impact”, and the roof structure of the car when it overturns during a "rollover
Rollover
A rollover is a type of vehicle accident in which a vehicle tips over onto its side or roof. The most common cause of a rollover is traveling too fast while turning.- Dynamics :Vehicles can roll over in several ways...
". Crash simulations can also be used to assess injury to pedestrians hit by a car.
Benefits
A crash simulation produces results without actual destructive testingDestructive testing
In destructive testing, tests are carried out to the specimen's failure, in order to understand a specimen's structural performance or material behaviour under different loads...
of a new car model. This way, tests can be performed quickly and inexpensively in a computer, which permits optimization of the design before a real prototype of the car has been manufactured. Using a simulation, problems can be solved before spending time and money on an actual crash test. The great flexibility of printed
Printing
Printing is a process for reproducing text and image, typically with ink on paper using a printing press. It is often carried out as a large-scale industrial process, and is an essential part of publishing and transaction printing....
output and graphical display enables designers to solve some problems that would have been nearly impossible without the help of a computer.
Analysis
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
to solve complex problems by dividing a surface into a large but still finite number of elements and determining the motion of these elements over very small periods of time. Essentially, using this method, one can solve many smaller problems instead of one larger one.
PAM-CRASH
PAM-CRASH
PAM-CRASH is a software package from ESI Group used for crash simulation and the design of occupant safety systems, primarily in the automotive industry. PAM-CRASH enables automotive engineers to simulate the performance of a proposed vehicle design and evaluate the potential for injury to vehicle...
started crash simulation and together with LS-DYNA
LS-DYNA
LS-DYNA is an advanced general-purpose multiphysics simulation software package developed by the Livermore Software Technology Corporation...
is a software package which is widely used for crash simulation.
Structural analysis
In a typical crash simulation, the car body structure is analyzed using spatial discretizationDiscretization
In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...
, that is, breaking up the continuous movement of the body in real time into smaller changes in position over small, discrete time steps. The discretization involves subdividing the surface of the constituent, thin, sheet metal
Sheet metal
Sheet metal is simply metal formed into thin and flat pieces. It is one of the fundamental forms used in metalworking, and can be cut and bent into a variety of different shapes. Countless everyday objects are constructed of the material...
parts into a large number (approaching one million in 2006) of quadrilateral
Quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
or triangular regions, each of which spans the area between "nodes" to which its corners are fixed. Each element has mass, which is distributed as concentrated masses and as mass moments of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
to its connecting nodes. Each node has 6 kinematic degrees of freedom
Degrees of freedom (engineering)
In mechanics, degrees of freedom are the set of independent displacements and/or rotations that specify completely the displaced or deformed position and orientation of the body or system...
, that is, one node can move in three linear directions under translation
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
and can rotate
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
about three independent axes. The spatial coordinates (x), displacement (u), velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
(v), and acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
(a) of each node is mostly expressed in a three-dimensional rectangular Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
with axes X,Y, and Z.
If the nodes move during a crash simulation, the connected elements move, stretch, and bend with their nodes, which causes them to impart forces and moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
to their nodal connections. The forces and moments at the nodes correspond to the inertia forces and moments, caused by their translational (linear) and angular acceleration
Angular acceleration
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha .- Mathematical definition :...
s and to the forces and moments transmitted by the resistance
Strength of materials
In materials science, the strength of a material is its ability to withstand an applied stress without failure. The applied stress may be tensile, compressive, or shear. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a...
of the structural material of the connected elements as they deform. Sometimes, additional external structural loads are applied, like gravity loads from the self weight of the parts, or added loads from external masses.
The forces and moments of all nodes are collected into a column vector (or column matrix), and the time dependent equations of motion
Equation of motion
Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...
(in dynamic equilibrium) can be written as follows.
where vector
(mass times acceleration vector) collects the inertia forces at the nodes, collects the external nodal loads, and collects the internal resisting forces from the deformation of the material. M is a diagonal matrixDiagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
of the nodal masses. Each vector (u, v, a, F, etc.) has dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
6 times the total number of nodes in the crash model (about 6 million “degrees of freedom
Degrees of freedom (engineering)
In mechanics, degrees of freedom are the set of independent displacements and/or rotations that specify completely the displaced or deformed position and orientation of the body or system...
” for every 1 million "nodes" in 3-D thin shell finite element models).
Time analysis
A crash simulation uses time discretization as well to separate the continuous changes in time into very small, usable segments. The dynamic equations of motion hold at all times during a crash simulation and must be integrated in time, t, starting from an initial condition at time zero, which is just prior to the crash. According to the explicit finite differenceFinite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
time integration method used by most crash codes, the accelerations, velocities, and displacements of the body are related by the following equations.
In these equations the subscripts n±1/2, n, n+1 denote past, present, and future times, t, at half and full time intervals with time steps
and , respectively.Solution
The above system of linear equations is solved for the accelerations,, the velocities, , and the displacements, , at each discrete point in time, t, during the crash duration. This solution is trivial, since the mass matrix is diagonal. The computer time is proportional to the number of finite elements and the number of solution time steps. The stable solution time step, , is limited for numerical stabilityNumerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
, as expressed by the Courant Fredericks Levy condition(CFL), which states that “in any time-marching computer simulation, the time step must be less than the time for some significant action to occur, and preferably considerably less." In a crash simulation, the fastest significant actions are the acoustic signals that travel inside the structural material.
The solid elastic stress wave speed amounts to
where
is the initial elastic modulus (before plastic deformation) of the material and is the mass density. The largest stable time step for a given material is therefore,where
is the smallest distance between any two nodes of the numerical crash simulation model.Since this distance can change during a simulation, the stable time step changes and must be updated continually as the solution proceeds in time. When using steel
Steel
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...
, the typical value of the stable time step is about one microsecond
Microsecond
A microsecond is an SI unit of time equal to one millionth of a second. Its symbol is µs.A microsecond is equal to 1000 nanoseconds or 1/1000 millisecond...
when the smallest discrete node distance in the mesh of the finite element model is about 5 millimeters. It needs then more than 100,000 time intervals to solve a crash event that lasts for one tenth of a second. This figure is exceeded in many industrial crash models demanding optimized crash solvers with High-Performance Computing (HPC
HPC
HPC may refer to:* Handheld PC* Hasty Pudding cipher* Health Professions Council* Hemangiopericytoma* Hematopoietic progenitor cell* High-performance computing** Windows HPC Server 2008, an operating system for high-performance computing by Microsoft...
) features, such as vectorization
Vectorization
Vectorization, in parallel computing, is a special case of parallelization, in which software programs that by default perform one operation at a time on a single thread are modified to perform multiple operations simultaneously....
and parallel computing
Parallel computing
Parallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...
.
See also
- Finite element method in structural mechanicsFinite element method in structural mechanicsThe Finite element method is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at...
- Finite element analysis
- Crash testCrash testA crash test is a form of destructive testing usually performed in order to ensure safe design standards in crashworthiness and crash compatibility for various modes of transportation or related systems and components.- Types :...