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97-469: LS-DYNA originated from the 3D FEA program DYNA3D , developed by Dr. John O. Hallquist at Lawrence Livermore National Laboratory (LLNL) in 1976. DYNA3D was created in order to simulate the impact of the Full Fuzing Option (FUFO) or " Dial-a-yield " nuclear bomb for low altitude release (impact velocity of ~ 40 m/s). At the time, no 3D software was available for simulating impact, and 2D software

194-532: A variational formulation , a discretization strategy, one or more solution algorithms, and post-processing procedures. Examples of the variational formulation are the Galerkin method , the discontinuous Galerkin method, mixed methods, etc. A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions), and (c)

291-406: A 2,000 kg (4,400 lb) pickup truck traveling 100 km/h (62 mph), colliding with the rail at a 25-degree angle. Flexible barriers include cable barriers and weak post corrugated guide rail systems. These are referred to as flexible barriers because they will deflect 1.6 to 2.6 m (5.2 to 8.5 ft) when struck by a typical passenger car or light truck. Impact energy

388-407: A black lid. Fitch barriers are often found in a triangular arrangement at the end of a guard rail between a highway and an exit lane (the area known as the gore ), along the most probable line of impact. The barriers in front contain the least sand, with each successive barrel containing more. When a vehicle collides with the barrels, the vehicle's kinetic energy is dissipated by the shattering of

485-409: A common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem: with a finite-dimensional version: where V {\displaystyle V} is a finite-dimensional subspace of H 0 1 {\displaystyle H_{0}^{1}} . There are many possible choices for V {\displaystyle V} (one possibility leads to

582-407: A continuous domain into a set of discrete sub-domains, usually called elements. Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations that arise from the problem of torsion of a cylinder . Courant's contribution was evolutionary, drawing on

679-452: A critical component of comprehensive security planning at nuclear facilities . The NRC's detailed guidelines on vehicle barriers demonstrate its commitment to maintaining high standards of safety and security at U.S. nuclear sites . Adherence to these regulations is crucial for mitigating risks associated with vehicle-based threats. Traffic barriers are categorized in two ways: by the function they serve, and by how much they deflect when

776-703: A form of Green's identities , we see that if u {\displaystyle u} solves P2, then we may define ϕ ( u , v ) {\displaystyle \phi (u,v)} for any v {\displaystyle v} by ∫ Ω f v d s = − ∫ Ω ∇ u ⋅ ∇ v d s ≡ − ϕ ( u , v ) , {\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),} where ∇ {\displaystyle \nabla } denotes

873-520: A formidable deterrent against potential threats, including vehicle-borne attacks and unauthorized access. Road blockers are equipped with mechanisms that allow for quick deployment and retraction when needed, providing a flexible and effective means of traffic control and security management. Platform barriers , Platform screen doors (PSDs) without the doors, are used when PSDs are not feasible due to cost, technological compatibility or other factors. Barriers are divided into three groups, based on

970-410: A greater threat to general health and well-being of the public than the obstacle it intends to protect. In many regions of the world, the concept of clear zone is taken into account when examining the distance of an obstacle or hazard from the edge of travelway. Clear zone , also known as clear recovery area or horizontal clearance is defined (through study) as a lateral distance in which a motorist on

1067-504: A large body of earlier results for PDEs developed by Lord Rayleigh , Walther Ritz , and Boris Galerkin . The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart , R. W. Clough with co-workers at UC Berkeley , O. C. Zienkiewicz with co-workers Ernest Hinton , Bruce Irons and others at Swansea University , Philippe G. Ciarlet at

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1164-494: A larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations . Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis ( FEA ). The subdivision of a whole domain into simpler parts has several advantages: Typical work out of the method involves: The global system of equations has known solution techniques and can be calculated from

1261-511: A license, and can also be used for viewing and postprocessing simulation results. Licensees of LS-DYNA automatically have access to all of the program's capabilities, from simple linear static mechanical analysis up to advanced thermal and flow solving methods. Furthermore, they have full use of LSTC's LS-OPT software, a standalone design optimization and probabilistic analysis package with an interface to LS-DYNA. LS-DYNA's potential applications are numerous and can be used in many fields. LS-DYNA

1358-420: A median and striking an oncoming vehicle in a head-on crash . Unlike roadside barriers, they must be designed to be struck from either side. Bridge barriers are designed to restrain vehicles from crashing off the side of a bridge and falling onto the roadway, river or railroad below. It is usually higher than roadside barrier, to prevent trucks, buses, pedestrians and cyclists from vaulting or rolling over

1455-410: A one-sided traffic barrier. Recycled tyres had been proposed for highway crash barriers by 2012, but many governments prefer sand-filled crash barriers because they have excellent energy-absorption characteristics and are easier to erect and dismantle. A Fitch Barrier is an energy-absorbing type of impact attenuator consisting of a group of sand-filled plastic barrels, usually yellow in color with

1552-402: A particular model class. Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the variational formulation and discretization strategy choices. Post-processing procedures are designed to extract the data of interest from a finite element solution. To meet

1649-466: A particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations . The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into

1746-514: A physical system with the underlying physics such as the Euler–Bernoulli beam equation , the heat equation , or the Navier-Stokes equations expressed in either PDE or integral equations , while the divided small elements of the complex problem represent different areas in the physical system. FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines) when

1843-462: A recoverable slope may travel outside of the travelway and return their vehicle safely to the roadway. This distance is commonly determined as the 85th percentile in a study comparable to the method of determining speed limits on roadways through speed studies and varies based on the classification of a roadway. In order to provide for adequate safety in roadside conditions, hazardous elements such as fixed obstacles or steep slopes can be placed outside of

1940-505: A set of functions of Ω {\displaystyle \Omega } . In the figure on the right, we have illustrated a triangulation of a 15-sided polygonal region Ω {\displaystyle \Omega } in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space V {\displaystyle V} would consist of functions that are linear on each triangle of

2037-420: A vehicle crashes into them. Roadside barriers are used to protect traffic from roadside obstacles or hazards, such as slopes steep enough to cause rollover crashes, fixed objects like bridge piers , and bodies of water. Roadside barriers can also be used with medians, to prevent vehicles from colliding with hazards within the median. Median barriers are used to prevent vehicles from crossing over

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2134-1217: Is 1 {\displaystyle 1} at x k {\displaystyle x_{k}} and zero at every x j , j ≠ k {\displaystyle x_{j},\;j\neq k} , i.e., v k ( x ) = { x − x k − 1 x k − x k − 1  if  x ∈ [ x k − 1 , x k ] , x k + 1 − x x k + 1 − x k  if  x ∈ [ x k , x k + 1 ] , 0  otherwise , {\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in [x_{k},x_{k+1}],\\0&{\text{ otherwise}},\end{cases}}} for k = 1 , … , n {\displaystyle k=1,\dots ,n} ; this basis

2231-426: Is a command shell, the appropriate executable for the computer's architecture, an input file, and enough free disk space to store the results. Input files use a simple ASCII format and thus can be prepared using any text editor . Many third-party simulation environments integrate some LS-DYNA preprocessing capabilities. LSTC also develops its own preprocessor, LS-PrePost , which is freely distributed, runs without

2328-411: Is a connected open region in the ( x , y ) {\displaystyle (x,y)} plane whose boundary ∂ Ω {\displaystyle \partial \Omega } is nice (e.g., a smooth manifold or a polygon ), and u x x {\displaystyle u_{xx}} and u y y {\displaystyle u_{yy}} denote

2425-456: Is a popular method for numerically solving differential equations arising in engineering and mathematical modeling . Typical problem areas of interest include the traditional fields of structural analysis , heat transfer , fluid flow , mass transport, and electromagnetic potential . Computers are usually used to perform the calculations required. With high-speed supercomputers , better solutions can be achieved, and are often required to solve

2522-413: Is a shifted and scaled tent function . For the two-dimensional case, we choose again one basis function v k {\displaystyle v_{k}} per vertex x k {\displaystyle x_{k}} of the triangulation of the planar region Ω {\displaystyle \Omega } . The function v k {\displaystyle v_{k}}

2619-725: Is also an inner product, this time on the Lp space L 2 ( 0 , 1 ) {\displaystyle L^{2}(0,1)} . An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique u {\displaystyle u} solving (2) and, therefore, P1. This solution is a-priori only a member of H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} , but using elliptic regularity, will be smooth if f {\displaystyle f} is. P1 and P2 are ready to be discretized, which leads to

2716-552: Is composed of steel-reinforced plastic boxes that are put in place where needed, linked together to form a longitudinal barrier, then ballasted with water. These have an advantage in that they can be assembled without heavy lifting equipment, but they cannot be used in freezing weather. Road blockers are used to enhance security by preventing unauthorized or hostile vehicles from entering sensitive or protected locations, such as government buildings, military installations, airports, embassies, and high-security facilities. They act as

2813-403: Is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}} where we define x 0 = 0 {\displaystyle x_{0}=0} and x n + 1 = 1 {\displaystyle x_{n+1}=1} . Observe that functions in V {\displaystyle V} are not differentiable according to

2910-415: Is dissipated through deformation of the rail elements, posts, soil and vehicle bodywork, and friction between the rail and vehicle. Box beam systems also spread the impact force over a number of posts due to the stiffness of the steel tube. Rigid barriers are usually constructed of reinforced concrete. A permanent concrete barrier will only deflect a negligible amount when struck by a vehicle. Instead,

3007-495: Is dissipated through tension in the rail elements, deformation of the rail elements, posts, soil and vehicle bodywork, and friction between the rail and vehicle. Semi-rigid barriers include box beam guide rail, heavy post blocked out corrugated guide rail and thrie-beam guide rail. Thrie-beam is similar to corrugated rail, but it has three ridges instead of two. They deflect 3 to 6 feet (0.91 to 1.83 m): more than rigid barriers, but less than flexible barriers. Impact energy

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3104-537: Is easier for twice continuously differentiable u {\displaystyle u} ( mean value theorem ) but may be proved in a distributional sense as well. We define a new operator or map ϕ ( u , v ) {\displaystyle \phi (u,v)} by using integration by parts on the right-hand-side of (1): where we have used the assumption that v ( 0 ) = v ( 1 ) = 0 {\displaystyle v(0)=v(1)=0} . If we integrate by parts using

3201-933: Is given, u {\displaystyle u} is an unknown function of x {\displaystyle x} , and u ″ {\displaystyle u''} is the second derivative of u {\displaystyle u} with respect to x {\displaystyle x} . P2 is a two-dimensional problem ( Dirichlet problem ) P2  : { u x x ( x , y ) + u y y ( x , y ) = f ( x , y )  in  Ω , u = 0  on  ∂ Ω , {\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega ,\\u=0&{\text{ on }}\partial \Omega ,\end{cases}}} where Ω {\displaystyle \Omega }

3298-507: Is not limited to any particular type of simulation. In a given simulation, any of LS-DYNA's many features can be combined to model a wide variety of physical events. However the main strength of the software lies in highly nonlinear simulations of high-speed events, preferably involving the deformation of sheet metal. (For example a car crashing into a traffic barrier .) Several variants of algorithms and multiphysics expansions were added to use these core capabilites in special fields. (For example

3395-427: Is not restricted to triangles (tetrahedra in 3-d or higher-order simplexes in multidimensional spaces). Still, it can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle). Examples of methods that use higher degree piecewise polynomial basis functions are

3492-508: Is sheetmetal forming. LS-DYNA accurately predicts the stresses and deformations experienced by the metal, and determines if the metal will fail. LS-DYNA supports adaptive remeshing and will refine the mesh during the analysis, as necessary, to increase accuracy and save time. Metal forming applications for LS-DYNA include: LS-DYNA is used by the aerospace industry to simulate bird strike , jet engine blade containment, and structural failure. Aerospace applications for LS-DYNA include: LS-DYNA

3589-789: Is that the inner products ⟨ v j , v k ⟩ = ∫ 0 1 v j v k d x {\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx} and ϕ ( v j , v k ) = ∫ 0 1 v j ′ v k ′ d x {\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx} will be zero for almost all j , k {\displaystyle j,k} . (The matrix containing ⟨ v j , v k ⟩ {\displaystyle \langle v_{j},v_{k}\rangle } in

3686-407: Is the unique function of V {\displaystyle V} whose value is 1 {\displaystyle 1} at x k {\displaystyle x_{k}} and zero at every x j , j ≠ k {\displaystyle x_{j},\;j\neq k} . Depending on the author, the word "element" in the "finite element method" refers to

3783-883: Is then implemented on a computer . The first step is to convert P1 and P2 into their equivalent weak formulations . If u {\displaystyle u} solves P1, then for any smooth function v {\displaystyle v} that satisfies the displacement boundary conditions, i.e. v = 0 {\displaystyle v=0} at x = 0 {\displaystyle x=0} and x = 1 {\displaystyle x=1} , we have Conversely, if u {\displaystyle u} with u ( 0 ) = u ( 1 ) = 0 {\displaystyle u(0)=u(1)=0} satisfies (1) for every smooth function v ( x ) {\displaystyle v(x)} then one may show that this u {\displaystyle u} will solve P1. The proof

3880-461: Is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all

3977-455: Is used extensively by researchers from military and defense. Some of these applications include: LS-DYNA is used in oil and gas industries to perform fatigue analysis on offshore structures, failure analysis of ships under the event of collision, and simulate fluid structure interactions. LS-DYNA applications for oil and gas industry include: Other LS-DYNA applications include: Finite element method The finite element method ( FEM )

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4074-864: The ( j , k ) {\displaystyle (j,k)} location is known as the Gramian matrix .) In the one dimensional case, the support of v k {\displaystyle v_{k}} is the interval [ x k − 1 , x k + 1 ] {\displaystyle [x_{k-1},x_{k+1}]} . Hence, the integrands of ⟨ v j , v k ⟩ {\displaystyle \langle v_{j},v_{k}\rangle } and ϕ ( v j , v k ) {\displaystyle \phi (v_{j},v_{k})} are identically zero whenever | j − k | > 1 {\displaystyle |j-k|>1} . Similarly, in

4171-483: The Runge-Kutta method . In step (2) above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system . The process is often carried out by FEM software using coordinate data generated from

4268-531: The deep drawing of steel sheets by electromagnetic forces or by explosives.) One example of a simulation that involved a unique combination of several features is the NASA JPL Mars Pathfinder landing, which simulated the gas and fabric of inflating airbags around the spaceship, and the subseqent impact and bouncing of the assembly on the martian soil. LS-DYNA's analysis capabilities: LS-DYNA's comprehensive library of material models: Some of

4365-841: The gradient and ⋅ {\displaystyle \cdot } denotes the dot product in the two-dimensional plane. Once more ϕ {\displaystyle \,\!\phi } can be turned into an inner product on a suitable space H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} of once differentiable functions of Ω {\displaystyle \Omega } that are zero on ∂ Ω {\displaystyle \partial \Omega } . We have also assumed that v ∈ H 0 1 ( Ω ) {\displaystyle v\in H_{0}^{1}(\Omega )} (see Sobolev spaces ). The existence and uniqueness of

4462-456: The hp-FEM and spectral FEM . More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques; the most popular are: The primary advantage of this choice of basis

4559-468: The initial values of the original problem to obtain a numerical answer. In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of Galerkin method . The process, in mathematical language,

4656-1094: The spectral method ). However, we take V {\displaystyle V} as a space of piecewise polynomial functions for the finite element method. We take the interval ( 0 , 1 ) {\displaystyle (0,1)} , choose n {\displaystyle n} values of x {\displaystyle x} with 0 = x 0 < x 1 < ⋯ < x n < x n + 1 = 1 {\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1} and we define V {\displaystyle V} by: V = { v : [ 0 , 1 ] → R : v  is continuous,  v | [ x k , x k + 1 ]  is linear for  k = 0 , … , n , and  v ( 0 ) = v ( 1 ) = 0 } {\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{

4753-450: The "gating" feature allows the vehicles to pass through the rail as it bends. If space allows, a guide rail may also be terminated by gradually curving it back to the point that the terminal is unlikely to be hit end-on, or, if possible, by embedding the end in a hillside or cut slope. An alternative to energy absorbing barrier terminals are impact attenuators . These are used for wider hazards that cannot be effectively protected with

4850-495: The University of Paris 6 and Richard Gallagher with co-workers at Cornell University . Further impetus was provided in these years by available open-source finite element programs. NASA sponsored the original version of NASTRAN . UC Berkeley made the finite element programs SAP IV and later OpenSees widely available. In Norway, the ship classification society Det Norske Veritas (now DNV GL ) developed Sesam in 1969 for use in

4947-516: The amount they deflect when struck by a vehicle and the mechanism the barrier uses to resist the impact forces. In the United States , traffic barriers are tested and classified according to the AASHTO Manual for Assessing Safety Hardware (MASH) standards, which recently superseded Federal Highway Administration NCHRP Report 350. Barrier deflections listed below are results from crash tests with

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5044-411: The analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Gilbert Strang and George Fix . The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism , heat transfer , and fluid dynamics . A finite element method is characterized by

5141-430: The angled lower section. For low-speed or low-angle impacts on these barriers, that may be sufficient to redirect the vehicle without damaging the bodywork. The disadvantage is there is a higher likelihood of rollover with a small car than the single slope or step barriers. Impact forces are resisted by a combination of the rigidity and mass of the barrier. Deflection is usually negligible. An early concrete barrier design

5238-478: The ban in 1998 to the entire National Highway System . To address the vaulting and rollover crashes, a new type of terminals were developed. The first generation of these terminals in the 1970s were breakaway cable terminals, in which the rail curves back on itself and is connected to a cable that runs between the first and second posts (which are often breakaway posts). These barrier terminals were sometimes able to spear through small cars that hit them at exactly

5335-429: The barrier and falling over the side of the structure. Bridge rails are usually multi-rail tubular steel barriers or reinforced concrete parapets and barriers. Work zone barriers are used to protect traffic from hazards in work zones. Their distinguishing feature is they can be relocated as conditions change in the road works. Two common types are used: temporary concrete barrier and water-filled barrier. The latter

5432-468: The barrier, potentially causing the vehicle to roll over. However, along parkways and other areas where aesthetics are considered important, reinforced concrete walls with stone veneers or faux stone finishes are sometimes used. These barrier walls usually have vertical faces to prevent vehicles from climbing the barrier. For several decades after the invention of motor vehicles, designers of early traffic barriers paid little attention to their ends, so that

5529-422: The barriers either ended abruptly in blunt ends, or sometimes featured some flaring of the edges away from the side of the barrier facing traffic. Vehicles that struck blunt ends at the wrong angle could stop too suddenly or suffer penetration of the passenger compartment by steel rail sections, resulting in severe injuries or fatalities. Traffic engineers have learned through such gruesome real-world experience that

5626-450: The chosen triangulation. One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter h > 0 {\displaystyle h>0} which one takes to be very small. This parameter will be related to

5723-402: The clear zone in order to reduce or eliminate the need for roadside protection. Common sites for installation of traffic barrier: When a barrier is needed, careful calculations are completed to determine length of need. The calculations take into account the speed and volume of traffic volume using the road, the distance from the edge of travelway to the hazard, and the distance or offset from

5820-451: The domain changes (as during a solid-state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations. For example, in a frontal crash simulation, it is possible to increase prediction accuracy in "important" areas like

5917-466: The domain's triangles, the piecewise linear basis function, or both. So, for instance, an author interested in curved domains might replace the triangles with curved primitives and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" with "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method

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6014-435: The edge of travelway to the barrier. In accordance with U.S. regulations for nuclear power plants , the U.S. Nuclear Regulatory Commission (NRC) addresses vehicle barriers under 10 CFR Part 73 , specifically in 10 CFR 73.55(e)(10) Vehicle Barriers. This section requires licensees to "use physical barriers and security strategies [via strategic planning ] to protect against land vehicle borne explosive devices ". Here,

6111-564: The element types available in LS-DYNA: LS-DYNA's contact algorithms: LS-DYNA is used by the automotive industry to analyze vehicle designs. LS-DYNA accurately predicts a car's behavior in a collision and the effects of the collision upon the car's occupants. With LS-DYNA, automotive companies and their suppliers can test car designs without having to tool or experimentally test a prototype, thus saving time and expense. LS-DYNA's specialized automotive features: One of LS-DYNA's applications

6208-611: The elementary definition of calculus. Indeed, if v ∈ V {\displaystyle v\in V} then the derivative is typically not defined at any x = x k {\displaystyle x=x_{k}} , k = 1 , … , n {\displaystyle k=1,\ldots ,n} . However, the derivative exists at every other value of x {\displaystyle x} , and one can use this derivative for integration by parts . We need V {\displaystyle V} to be

6305-603: The ends of barriers are just as important as the barriers themselves; the American Association of State Highway and Transportation Officials devotes an entire chapter to the topic of barrier "end treatments" in its Roadsign Design Guide . In response, a new style of barrier terminals was developed in the 1960s in which the installers were directed to twist the guardrail 90 degrees and bring its end down so that it would lie flat at ground level (so-called "turned-down" terminals or "ramped ends"). While this innovation prevented

6402-473: The execution speed by about 10 percent. Hallquist was the sole developer of DYNA3D until 1984, when he was joined by Dr. David J. Benson. In 1986, many capabilities were added. The added features included beams, shells, rigid bodies, single surface contact, interface friction, discrete springs and dampers, optional hourglass treatments, optional exact volume integration, and VAX / VMS , IBM , UNIX , COS operating system compatibility. At this point, DYNA3D became

6499-455: The finite element method for P1 and outline its generalization to P2. Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM. After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem

6596-464: The finite element method. P1 is a one-dimensional problem  P1  : { u ″ ( x ) = f ( x )  in  ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , {\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}} where f {\displaystyle f}

6693-476: The first code to have a general single surface contact algorithm. Metal forming simulation and composite analysis capabilities were added to DYNA3D in 1987. This version included changes to the shell elements, and dynamic relaxation . The final release of DYNA3D in 1988 included several more elements and capabilities. By 1988 LLNL had sent approximately 600 tapes containing simulation software. Hallquist had consulted for nearly 60 companies and organizations on

6790-402: The first version, while including element-wise integration of the integral difference method developed in 1974. The 1982 release included nine additional material models which allowed for new simulations, such as explosive-structure and soil-structure interactions. The release also permitted the analysis of structural response due to penetrating projectiles . Improvements in 1982 further boosted

6887-573: The focus is on safeguarding the protected area and vital areas of nuclear facilities from unauthorized vehicle access, emphasizing the need for effective barrier systems against potential vehicular threats. The regulation highlights the importance of designing and implementing barriers that are robust enough to withstand various threat scenarios, including different types of vehicles and potential explosive devices . The integration of these barriers with other security measures, such as surveillance , access control , and intrusion detection systems , forms

6984-439: The front of the car and reduce it in its rear (thus reducing the cost of the simulation). Another example would be in numerical weather prediction , where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas. A clear, detailed, and practical presentation of this approach can be found in

7081-399: The largest and most complex problems. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems ). There are also studies about using FEM solve high-dimensional problems. To solve a problem, the FEM subdivides a large system into smaller, simpler parts called finite elements . This is achieved by

7178-401: The largest or average triangle size in the triangulation. As we refine the triangulation, the space of piecewise linear functions V {\displaystyle V} must also change with h {\displaystyle h} . For this reason, one often reads V h {\displaystyle V_{h}} instead of V {\displaystyle V} in

7275-481: The literature. Since we do not perform such an analysis, we will not use this notation. To complete the discretization, we must select a basis of V {\displaystyle V} . In the one-dimensional case, for each control point x k {\displaystyle x_{k}} we will choose the piecewise linear function v k {\displaystyle v_{k}} in V {\displaystyle V} whose value

7372-411: The mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, p-version , hp-version , x-FEM , isogeometric analysis , etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in

7469-414: The opposing carriageway of traffic and help to reduce head-on collisions . Some of these barriers, designed to be struck from either side, are called median barriers. Traffic barriers can also be used to protect vulnerable areas like school yards, pedestrian zones , and fuel tanks from errant vehicles. In pedestrian zones, like school yards, they also prevent children or other pedestrians from running onto

7566-417: The performance limits of traffic barriers and provide an adequate level of protection to road users. Roadside hazards must be assessed for the danger they pose to traveling motorists based on size, shape, rigidity, and distance from the edge of travelway. For instance, small roadside signs and some large signs (ground-mounted breakaway post) often do not merit roadside protection as the barrier itself may pose

7663-1902: The planar case, if x j {\displaystyle x_{j}} and x k {\displaystyle x_{k}} do not share an edge of the triangulation, then the integrals ∫ Ω v j v k d s {\displaystyle \int _{\Omega }v_{j}v_{k}\,ds} and ∫ Ω ∇ v j ⋅ ∇ v k d s {\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds} are both zero. If we write u ( x ) = ∑ k = 1 n u k v k ( x ) {\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)} and f ( x ) = ∑ k = 1 n f k v k ( x ) {\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)} then problem (3), taking v ( x ) = v j ( x ) {\displaystyle v(x)=v_{j}(x)} for j = 1 , … , n {\displaystyle j=1,\dots ,n} , becomes Traffic barrier Traffic barriers (known in North America as guardrails or guard rails , in Britain as crash barriers , and in auto racing as Armco barriers ) keep vehicles within their roadway and prevent them from colliding with dangerous obstacles such as boulders, sign supports, trees, bridge abutments, buildings, walls, and large storm drains , or from traversing steep (non-recoverable) slopes or entering deep water. They are also installed within medians of divided highways to prevent errant vehicles from entering

7760-540: The rail from penetrating the vehicle, it could also vault a vehicle into the air or cause it to roll over, since the rising and twisting guardrail formed a ramp. These crashes often led to vehicles vaulting, rolling, or vaulting and rolling at high speed into the very objects which guardrails or barriers were supposed to protect them from in the first place. Such wild crashes caused the United States to ban ramped ends in 1990 on high-speed, high-volume highways, and to extend

7857-471: The requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. Some very efficient postprocessors provide for the realization of superconvergence . The following two problems demonstrate

7954-671: The road. While barriers are normally designed to minimize injury to vehicle occupants, injuries do occur in collisions with traffic barriers. They should only be installed where a collision with the barrier is likely to be less severe than a collision with the hazard behind it. Where possible, it is preferable to remove, relocate or modify a hazard, rather than shield it with a barrier. To make sure they are safe and effective, traffic barriers undergo extensive simulated and full scale crash testing before they are approved for general use. While crash testing cannot replicate every potential manner of impact, testing programs are designed to determine

8051-602: The same software was developed concurrently. In 1978 the DYNA3D source code was released into the public domain without restrictions after a request from France. In 1979 a new version of DYNA3D was released which was programmed for optimal performance on the CRAY-1 supercomputers. This new release contained improved sliding interface treatment which was an order of magnitude faster than the previous contact treatment. This version also eliminated structural and higher order solid elements of

8148-529: The second derivatives with respect to x {\displaystyle x} and y {\displaystyle y} , respectively. The problem P1 can be solved directly by computing antiderivatives . However, this method of solving the boundary value problem (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like u + V ″ = f {\displaystyle u+V''=f} . For this reason, we will develop

8245-414: The shape of a concrete barrier is designed to redirect a vehicle into a path parallel to the barrier. This means they can be used to protect traffic from hazards very close behind the barrier, and generally require very little maintenance. Impact energy is dissipated through redirection and deformation of the vehicle itself. Jersey barriers and F-shape barriers also lift the vehicle as the tires ride up on

8342-553: The solution can also be shown. We can loosely think of H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} to be the absolutely continuous functions of ( 0 , 1 ) {\displaystyle (0,1)} that are 0 {\displaystyle 0} at x = 0 {\displaystyle x=0} and x = 1 {\displaystyle x=1} (see Sobolev spaces ). Such functions are (weakly) once differentiable, and it turns out that

8439-484: The spatial derivatives from the PDE, thus approximating the PDE locally with These equation sets are element equations. They are linear if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using numerical linear algebra methods. In contrast, ordinary differential equation sets that occur in the transient problems are solved by numerical integration using standard techniques such as Euler's method or

8536-404: The subdomains. The practical application of FEM is known as finite element analysis (FEA). FEA as applied in engineering , is a computational tool for performing engineering analysis . It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually

8633-479: The symmetric bilinear map ϕ {\displaystyle \!\,\phi } then defines an inner product which turns H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side ∫ 0 1 f ( x ) v ( x ) d x {\displaystyle \int _{0}^{1}f(x)v(x)dx}

8730-419: The textbook The Finite Element Method for Engineers . While it is difficult to quote the date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering . Its development can be traced back to work by Alexander Hrennikoff and Richard Courant in the early 1940s. Another pioneer

8827-485: The use of DYNA3D. As a result, at the end of 1988 Livermore Software Technology Corporation (LSTC) was founded to continue the development of DYNA3D in a much more focused manner, resulting in LS-DYNA3D (later shortened to LS-DYNA). Releases and support for DYNA3D were thus halted. Since then, LSTC has greatly expanded the capabilities of LS-DYNA in an attempt to create a universal tool for most simulation needs. In 2019, LSTC

8924-520: The wrong angle and were deprecated in 1993. The second generation of these terminals, called energy-absorbing terminals, was developed in the 1990s and 2000s. The goal was to develop a kinetic energy dissipating system soft enough for small vehicles to decelerate without causing the guardrail to spear through them, but firm enough to stop larger vehicles. The energy dissipation could be done through bending, kinking, crushing, or deforming guardrail elements. The first family of energy-absorbing terminal products

9021-608: Was Ioannis Argyris . In the USSR, the introduction of the practical application of the method is usually connected with the name of Leonard Oganesyan . It was also independently rediscovered in China by Feng Kang in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the finite difference method based on variation principle . Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of

9118-417: Was acquired by Ansys, Inc . Nonlinear means at least one (and sometimes all) of the following complications: Transient dynamic means analyzing high speed, short duration events where inertial forces are important. Typical uses include: LS-DYNA consists of a single executable file and is entirely command-line driven. Therefore, all that is required to run LS-DYNA (besides some licensing infrastructure)

9215-522: Was developed by the New Jersey State Highway Department. This led to the term Jersey barrier being used as a generic term, although technically it applies to a specific shape of concrete barrier. Other types include constant-slope barriers , concrete step barriers , and F-shape barriers . Concrete barriers usually have smooth finishes. At some impact angles, coarse finishes allow the drive wheel of front wheel drive vehicles to climb

9312-467: Was inadequate. Though the FUFO bomb was eventually canceled, development of DYNA3D continued. DYNA3D used explicit time integration to study nonlinear dynamic problems, with the original applications being mostly stress analysis of structures undergoing various types of impacts. The program was initially very simple largely due to the lack of adequate computational resources at the time. A two-dimensional version of

9409-410: Was the extruding terminal type. It features a large steel impact head that engages the frame or bumper of the vehicle in head-on collisions. The impact head is driven back along the guide rail, dissipating the vehicle's kinetic energy by bending or tearing the steel in the guide rail sections away to the side to prevent spearing. When the terminals are hit in an angle, they dissipate much of the energy but

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