Classifications

• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F30/00—Computer-aided design [CAD]
  • G06F30/20—Design optimisation, verification or simulation
• • G—PHYSICS
  • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
  • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
  • G16C10/00—Computational theoretical chemistry, i.e. ICT specially adapted for theoretical aspects of quantum chemistry, molecular mechanics, molecular dynamics or the like

Definitions

• the present invention relates to a method of simulating a set of elements, according to which the behavior of the elements is determined, at successive simulation steps, on the basis of a Hamiltonian associated with the element system (the sum of the kinetic energy and potential energy of the whole)
• H - ⁇ p T ⁇ .M + V p, where p is a vector indicating elements the moments, V is the potential energy of the system and M "1 of a diagonal matrix depending on the masses of the elements (in the case, this matrix may be a function of the positions of the elements).
• the potential energy V is, in some cases, a function of the positions of the elements only. In other cases, the potential energy V may also depend on the moments of the elements. The forces acting on the elements can be derived from this potential energy.
• the simulation of a set of elements makes it possible to study the behavior of such a set and to analyze its properties: displacements in terms of positions and successive moments of the elements, correlations of displacements between elements, changes of structure, the increases and decreases of interactions between elements, the configurations adopted on average, the evolutions of the associated energies, etc.
• the elements may represent mechanical bodies, for example celestial or fluid, particles such as atoms or molecules, for example proteins, fluids, etc.
• a common way of simulating a set of elements is to consider the Hamiltonian of the set, to derive motion equations, and to deduce the motion of the elements according to these equations.
• WO 2009/007550 describes for example a simulation technique of a set of elements.
• the present invention aims to propose a new solution to reduce these problems.
• the invention proposes a method for simulating a set of elements of the aforementioned type, characterized in that said method comprises the steps according to which:
• the system of elements is represented in a k-ary tree comprising leaf nodes, said leaf nodes each representing a respective element, and internal nodes including a root node
• R is the root node of the tree, given the recursive formula according to which, for any node A of the tree having k child nodes A Y , A 2 , A k ,
• ⁇ ⁇ ⁇ PA -r ⁇ , - + r (i1 - ⁇ A ⁇ )) ⁇ 0 ' . 0
• the matrix E being a matrix of size dn A * dn A formed of n A * n A blocks of size d * d equal to the identity matrix of dimension d, n A being equal to the number of elements descending from the node A, d being the dimension of the space in which the particles evolve, m A is the sum of the mass of these n A falling elements of the node A, p A being a function of restriction of the node A between 0 and 1, whose value is equal to 1 when A is a leaf node or when the same movement has been imposed on the descending elements of the given node; in the case of a leaf node A i , the matrix
• ⁇ ⁇ is equal to the inverse mass of the particle represented by the node A i multiplied by the identity matrix of dimension d.
• the invention makes it possible to perform simulations that require a smaller calculation volume and, consequently, less computing time, to determine the behavior of the elements according to these simulations, for example the potential energy, the forces applied to the elements, positions and / or moments of the elements.
• the method of simulating a set of elements according to the invention further comprises one or more of the following features: for the current simulation step, the same translational movement is imposed on at least the descending elements of a given node of the tree as a function of the value of a function of the moments of said elements; for the current simulation step, the same translational movement is imposed on at least the descending elements of a given node of the tree as a function of the value taken by, or
• m A is the sum of the mass of the elements of node A ;
• p A s i is the sum vector of the moments of the elements of the child node A ⁇ , ⁇ .
• ⁇ is the norm of the vector, C is a positive constant;
• p A 1 is fixed to impose the same translational movement on the descending elements of the node A;
• the method comprises a step of determining the values of at least one piece of information, at successive simulation instants on the basis of said Hamiltonian, said stage taking advantage of the fact that the values of the information relating to the elements to which the same movement in translation has been imposed for the moment of current simulation, depend on the relative position of said elements and are therefore unchanged;
• the information relating to said element comprises the potential energy of said element and / or the interaction force applied to said element.
• the present invention provides a computer program for simulating a system of elements, comprising software instructions for implementing the steps of a method according to the first aspect of the invention in a execution of the program by calculation means.
• FIG. 1 represents a simulation of trajectories of a 2-particle system performed with the standard Hamiltonian H;
• FIG. 2 represents trajectory simulations of a 2-particle system with the relative adaptive Hamiltonian H A R according to the invention
• FIG. 3 represents a device embodying an embodiment of the invention
• FIG. 4 is a flowchart of the steps of a method in one embodiment of the invention.
• FIG. 5 represents a flowchart of the substeps of step 101 of FIG. 4.
• V (q) is the interaction potential between the N particles; it is a function of their positions and it will be considered as independent of moments.
• a Hamiltonian said Hamiltonian, is defined that is adaptive relative to H AR , thus:
• H AR (p, q) -p 1 & (p, q) .p + V (q), (formula 1) in which ( ⁇ , ⁇ ), 3N * 3N diagonal block matrix referred to as inverse adaptive mass matrix relative, replaces M "1 and depends on the vector p, and optionally on the vector q.
• the values taken by ⁇ ( ⁇ , ⁇ ) are determined at each simulation step, the function ⁇ having been chosen so as to be able to restrict at least a subset of the particles to follow together the same movement with one or some no simulation.
• the particles that are part of the subset are determined by verification of a condition, for example to verify that before the current simulation step, they were driven by a close movement.
• the particles of such a subset can thus be assimilated, for the simulation step considered, as part of a single rigid body.
• the information that depends only on the relative positions of the particles, for example, the interaction forces between the particles, does not need to be updated at the end of the simulation step considered within the same sub-group. rigid set.
• each particle resumes its own motion.
• a continuous transition is applied between the two types of behaviors (restricted movement / free movement).
• H A R adaptive Hamiltonian
• NVE set E particle number, constant volume and temperature
• a first solution is to impose constraints on determined pairs of rigid bodies that can be considered as candidates to merge into a single rigid body. For example, noting A 1; At n , a set of rigid indexed bodies, the indices can be organized into a binary tree, so that it can only merge with A 2 into a rigid body (A ! + A 2 ), when a specific fusion condition is satisfied, A 3 can only merge with A 4 etc., and at a higher level in the binary tree, (A ! + A 2 ) can only merge with (A 3 + A 4 ) etc.
• Such a tree comprises at the maximum hierarchical rank, a root node, named node R, and at the minimum hierarchical rank, the leaf nodes corresponding to the particles considered alone. All the nodes of the tree, except the leaf nodes, are called the internal nodes.
• Another solution is to impose constraints on determined subsets of rigid bodies, possibly comprising more than two rigid bodies. We could thus organize indices into an n-ary tree, so that Ai can merge only with A 2 , A 3 , ..., A n at the same time, and so on.
• Another solution is to consider the graph derived from a search of the neighbors, the nodes of the graph being particles or rigid bodies, and an edge between two nodes indicates that the distance between the two nodes connected by the edge is less than a threshold distance.
• the embodiment detailed below is based on a binary tree structure as mentioned with respect to the first solution; the tree structure has the advantage of allowing incremental and recursive updates of the decision metrics for the internal nodes in the hierarchy, and incremental updates, for example for the forces and partial energies.
• each leaf node represents a single particle of the set
• each inner node represents a subset of particles composed of particles represented by the child nodes of said internal node.
• C will be called the set comprising node C and the descendants, direct or indirect, of node C.
• ⁇ 0 corresponding to this node C with the following recursive formula:
• the adaptive inverse mass matrix ⁇ ⁇ corresponding to the node A and the adaptive inverse mass matrix ⁇ ⁇ corresponding to the node B are themselves defined according to the formula 3 if the nodes A and B are not leaf nodes; in the case of a leaf node A, respectively B, the matrix ⁇ ⁇ , respectively ⁇ ⁇ , is equal to the inverse mass of the particle represented by the node A, respectively the node B, multiplied by the identity matrix of dimension 3 ;
• the matrix E is a matrix of size 3n c * 3n c , formed of n c * n c blocks 3 * 3 equal to the identity matrix of dimension 3, n c being equal to the number of leaf nodes of the node C (ie set of leaf nodes in the direct child nodes of node C or in the descendants of these child nodes);
• p c is the restriction function of body C.
• All the leaf nodes are considered rigid by definition, and this all the time that the simulation lasts, since they consist each of a single particle.
• the function p c is defined recursively, so as to smooth the changeover between the values 0 and 1, according to the following formula:
• ⁇ ( ⁇ ) is an order interpolation function
• ⁇ 0 is chosen depending on a relation of the moment of the two bodies A and B, with
• This invention can be generalized for a k-ary tree.
• the parent node with k child nodes A 1 ⁇ i k k the inverse matrix of the inertia is
• the coefficient 1 ⁇ 2 in this formula and the formula 6 can be replaced by another constant, for example 1, which will not change the simulation if the thresholds ⁇ 'and ⁇ ⁇ are modified in the same way, ie multiplied by an equal factor double the other constant).
• This invention can be generalized as differently: one can choose the restriction function p c different for different internal nodes. For example, up to a certain level of the tree p c is defined according to formulas 4 and 5, therefore, the nodes can be joined together and separated, and in the levels of the tree above this level, p c is always equal to zero.
• it is possible to define a subset of the system E which is permanently active the particles of this subset are never merged with each other or with the rest of the system, so they always follow the During the simulation, an unrestricted, free movement (such an embodiment of the invention may for example be applied to a polymer in a solvent): all the particles of this subset must be placed in a separate subtree.
• a one-dimensional system comprising two particles P1, P2 of mass 1 connected to one another by a spring of stiffness 1.
• the binary tree corresponding to the system is a root node C whose two child nodes A and B correspond to these two particles. Given initial moments, the two particles oscillate in space with time.
• FIG. 1 The space-time trajectories calculated for these particles according to a simulation of the prior art based on the standard Hamiltonian H are shown in FIG. 1 as a function of the simulation time.
• the trajectories d calculated for these particles according to a simulation on the basis of the relative adaptive Hamiltonian H A R according to the invention are represented in FIG. 2 as a function of the simulation time, for different values taken by ⁇ and ⁇ f .
• the moments of the two particles become close according to the formula 6. This occurs when the spring is almost compressed and almost decompressed.
• the root node C then becomes rigid. Consequently, in a general case, the son nodes of the rigid node C follow the same translational movement, in this case, because of the preservation of the moments, the particles stop.
• the trajectories 2 particles therefore become parallel.
• the moments of the particles continue to evolve however, which has the consequence that at a given pitch of the simulation, the rigidity condition of the node C is no longer satisfied and that the two particles resume their own motion. This is repeated periodically during the simulation.
• the transition region of the trajectory is wider.
• the stiffening region is longer and flatter.
• the matrix ⁇ defined as ⁇ 3 ⁇ 4, specifies how, and when, degrees of freedom in the relative position of the subset (s) of particles are activated or deactivated during the simulation.
• a computer device 1 shown in FIG. 3 is used to implement a simulation of a set E of N particles, according to the principles of the invention explained above.
• This device 1 comprises a computer including in particular a memory 2 adapted to store software programs and calculated parameter values successively described below (global interaction forces, partial, interaction potential, positions, moments ...), a microprocessor 3 adapted to execute the software program instructions and in particular the program P described below, and a man / machine interface 4, comprising for example a keyboard and a screen, respectively to enter instructions from a user and to display information intended for the user, for example curves such as those illustrated in FIG. 2.
• a computer including in particular a memory 2 adapted to store software programs and calculated parameter values successively described below (global interaction forces, partial, interaction potential, positions, moments ...), a microprocessor 3 adapted to execute the software program instructions and in particular the program P described below, and a man / machine interface 4, comprising for example a keyboard and a screen, respectively to enter instructions from a user and to display information intended for the user, for example curves such as those illustrated in FIG. 2.
• a binary tree representing the particles and the possibilities of stiffening two in two, as indicated above, is built for system E, initial moment and position values are set for the particles. Only the leaf nodes are then rigid.
• a step 101 the interaction forces exerted between the particles at the n + 1 ' th iteration, are determined.
• a step 102 the moments of the particles at the n + 1 ' th iteration, are determined.
• the metrics relating to the internal nodes of the tree at the n + 1 ' th iteration are determined. They include in particular p c and e c .
• a step 104 the positions of the particles at the n + 1 ' th iteration, are determined.
• step 101 the updating of the value of the interaction forces can be accelerated, compared to the prior art, taking into account the stiffenings according to the invention, if the interaction forces depend solely on the positions related particles.
• This algorithm is based, first, on a hierarchy of bounding volumes
• a partial force table corresponding to each internal node C comprises 3D vectors, the number of which is equal to the number of leaf nodes descending directly or indirectly from the node C. Each vector of this table is the interaction force acting on the particle represented by one of these leaf nodes in from all the other particles represented by the other leaf nodes descending from C. Each vector of the partial force table of the root node R corresponds to the global force acting on a particle.
• a partial force table corresponding to a leaf node is a null vector.
• the force update at each iteration includes three main substeps (the potential energy can be updated at the same time):
• the bounding volume hierarchy is updated, for example from the bottom of the tree upward (i.e. from the leaf nodes to the root).
• the updated AABB box coincides with the last determined position of the particle represented by the leaf node.
• the box of each internal node is recalculated for example by including the boxes of its two child nodes.
• the structure of the tree is unchanged, the surrounding volumes are updated according to the positions of the particles that have changed meanwhile.
• interaction lists are built recursively for the inner nodes of the tree.
• the interaction list thus constructed contains all the pairs of particles such that one of the particles of the pair belongs to the body A (ie is represented by a falling leaf node directly or indirectly from node A), while the other belongs to body B.
• This interaction list can be constructed for each inner node as described for other types of bounding volumes in R. Rossi, M. Isorce, S. Morin, J. Flocard, K.
• the interaction lists are built for all internal nodes. For all the other iterations, in the rigid nodes these lists do not change and therefore can not be recalculated.
• a step 101_3 an incremental update of the partial force tables is performed.
• a node is identified as a rigid node, the forces between the particles of the node have not changed. Therefore, only the partial force tables of the non-rigid nodes must be updated.
• They can be updated for example as follows recursively, starting from the leaf nodes to the root node.
• the elements of the partial force table relating to node A are copied and in the second half of the partial force table relative to node C, the elements of the partial force table relating to the node B are copied.
• the interaction forces between the two particles of each pair in this list are calculated and added to the corresponding forces of the force table. partial.
• the forces taking into account all the particles are stored in the partial force table of the root node R.
• the steps 101_2 and 101_3 can be combined, for example in a single traversal of the tree, to update the partial force table corresponding to a node C since the list of interactions corresponding to said node C is available.
• the other information depending only on the relative positions of the particles can be updated in a similar way.
• the step 102 of updating the moments of the particles is carried out using a conventional method.
• the complexity can be reduced to O (N.logN) by a recursive update from leaf nodes of data structures Q c , Rc and S c for each internal node C with two child nodes A and B.
• ⁇ x ⁇ is the dimension vector equal to n c and each of the n c components is equal to the element x.
• the function ( ⁇ ) represents the dot product.
• step 103 includes updating the structures Q c , R c and S c for each node C comprising two child nodes A and B.
• step 103 the update metric Q c, R c and S c and e c and p c with respect to the node C is performed recursively along the nodes of the hierarchy from the root:
• the update of the vectors Q c , Rc and S c can be carried out only for the nodes for which p c > 0 because these metrics may not be used for free nodes, so there is no need to update them.
• these metrics can be updated other than by traversing the tree, as long as they are updated in the subtrees with the rigid node at the top.
• the positions of the particles are updated in step 104.
• step 104 may be the following for updating the position of a node C, each internal node C being considered as having two child nodes A and B:
• the position of the particles is thus determined.
• the one-to-one correspondence between the identifier of the particle i and its serial number among the child nodes of the node C must be established, for example during the initialization of the simulation, for all the particles and all the nodes. internal of the tree.
• the metrics of the nodes for which p c > 0 are not used and therefore may not be calculated. If all the metrics are still updated during step 103, the positions of all the particles t , l ⁇ i ⁇ N can also be determined as follows:
• the positions can be updated other than by traversing the tree.
• q is the coordinate vector of the particle a, at the last step of the adaptive simulation
• qf is the coordinate vector of the same particle at the last step of the reference simulation.
• a method according to the invention makes it possible to accelerate the calculations, with a potentially low alteration of the behaviors.
• the fusion of two subsets into a rigid set has been considered, in other embodiments, the number of subsets merged into a rigid set is higher than together.
• the transition region between the two rigid and free states can be taken of greater or smaller width.

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