| [d82961] | 1 | /* | 
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|  | 2 | * Project: MoleCuilder | 
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|  | 3 | * Description: creates and alters molecular systems | 
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|  | 4 | * Copyright (C)  2011 University of Bonn. All rights reserved. | 
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|  | 5 | * Please see the LICENSE file or "Copyright notice" in builder.cpp for details. | 
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|  | 6 | */ | 
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|  | 7 |  | 
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|  | 8 | /* | 
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|  | 9 | * LinkedCell_Model.cpp | 
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|  | 10 | * | 
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|  | 11 | *  Created on: Nov 15, 2011 | 
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|  | 12 | *      Author: heber | 
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|  | 13 | */ | 
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|  | 14 |  | 
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|  | 15 | // include config.h | 
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|  | 16 | #ifdef HAVE_CONFIG_H | 
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|  | 17 | #include <config.h> | 
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|  | 18 | #endif | 
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|  | 19 |  | 
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|  | 20 | #include "CodePatterns/MemDebug.hpp" | 
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|  | 21 |  | 
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|  | 22 | #include "LinkedCell_Model.hpp" | 
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|  | 23 |  | 
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|  | 24 | #include <algorithm> | 
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|  | 25 | #include <boost/multi_array.hpp> | 
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|  | 26 | #include <limits> | 
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|  | 27 |  | 
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| [402f2c] | 28 | #include "Atom/AtomObserver.hpp" | 
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|  | 29 | #include "Atom/TesselPoint.hpp" | 
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| [d82961] | 30 | #include "Box.hpp" | 
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|  | 31 | #include "CodePatterns/Assert.hpp" | 
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|  | 32 | #include "CodePatterns/Info.hpp" | 
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|  | 33 | #include "CodePatterns/Log.hpp" | 
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| [2614e2a] | 34 | #include "CodePatterns/Observer/Observer.hpp" | 
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|  | 35 | #include "CodePatterns/toString.hpp" | 
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| [d82961] | 36 | #include "LinearAlgebra/RealSpaceMatrix.hpp" | 
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|  | 37 | #include "LinearAlgebra/Vector.hpp" | 
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|  | 38 | #include "LinkedCell/IPointCloud.hpp" | 
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|  | 39 | #include "LinkedCell/LinkedCell.hpp" | 
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| [402f2c] | 40 | #include "World.hpp" | 
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| [d82961] | 41 |  | 
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|  | 42 | #include "LinkedCell_Model_inline.hpp" | 
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|  | 43 |  | 
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|  | 44 | // initialize static entities | 
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|  | 45 | LinkedCell::tripleIndex LinkedCell::LinkedCell_Model::NearestNeighbors; | 
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|  | 46 |  | 
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|  | 47 | /** Constructor of LinkedCell_Model. | 
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|  | 48 | * | 
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|  | 49 | * @param radius desired maximum neighborhood distance | 
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|  | 50 | * @param _domain Box instance with domain size and boundary conditions | 
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|  | 51 | */ | 
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|  | 52 | LinkedCell::LinkedCell_Model::LinkedCell_Model(const double radius, const Box &_domain) : | 
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| [2614e2a] | 53 | ::Observer(std::string("LinkedCell_Model")+std::string("_")+toString(radius)), | 
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| [d82961] | 54 | internal_Sizes(NULL), | 
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|  | 55 | domain(_domain) | 
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|  | 56 | { | 
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|  | 57 | // set default argument | 
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|  | 58 | NearestNeighbors[0] = NearestNeighbors[1] = NearestNeighbors[2] = 1; | 
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|  | 59 |  | 
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| [2614e2a] | 60 | // get the partition of the domain | 
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| [d82961] | 61 | setPartition(radius); | 
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|  | 62 |  | 
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|  | 63 | // allocate linked cell structure | 
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|  | 64 | AllocateCells(); | 
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| [402f2c] | 65 |  | 
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|  | 66 | // sign in to AtomObserver | 
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|  | 67 | startListening(); | 
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| [d82961] | 68 | } | 
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|  | 69 |  | 
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|  | 70 | /** Constructor of LinkedCell_Model. | 
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|  | 71 | * | 
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|  | 72 | * @oaram set set of points to place into the linked cell structure | 
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|  | 73 | * @param radius desired maximum neighborhood distance | 
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|  | 74 | * @param _domain Box instance with domain size and boundary conditions | 
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|  | 75 | */ | 
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|  | 76 | LinkedCell::LinkedCell_Model::LinkedCell_Model(IPointCloud &set, const double radius, const Box &_domain) : | 
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| [2614e2a] | 77 | ::Observer(std::string("LinkedCell_Model")+std::string("_")+toString(radius)), | 
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| [d82961] | 78 | internal_Sizes(NULL), | 
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|  | 79 | domain(_domain) | 
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|  | 80 | { | 
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|  | 81 | Info info(__func__); | 
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|  | 82 |  | 
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|  | 83 | // get the partition of the domain | 
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|  | 84 | setPartition(radius); | 
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|  | 85 |  | 
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|  | 86 | // allocate linked cell structure | 
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|  | 87 | AllocateCells(); | 
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|  | 88 |  | 
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|  | 89 | insertPointCloud(set); | 
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| [402f2c] | 90 |  | 
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|  | 91 | // sign in to AtomObserver | 
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|  | 92 | startListening(); | 
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| [d82961] | 93 | } | 
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|  | 94 |  | 
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|  | 95 | /** Destructor of class LinkedCell_Model. | 
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|  | 96 | * | 
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|  | 97 | */ | 
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|  | 98 | LinkedCell::LinkedCell_Model::~LinkedCell_Model() | 
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|  | 99 | { | 
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| [402f2c] | 100 | // sign off from observables | 
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|  | 101 | stopListening(); | 
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|  | 102 |  | 
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|  | 103 | // reset linked cell structure | 
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| [d82961] | 104 | Reset(); | 
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|  | 105 | } | 
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|  | 106 |  | 
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| [402f2c] | 107 | /** Signs in to AtomObserver and World to known about all changes. | 
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|  | 108 | * | 
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|  | 109 | */ | 
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|  | 110 | void LinkedCell::LinkedCell_Model::startListening() | 
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|  | 111 | { | 
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|  | 112 | World::getInstance().signOn(this, World::AtomInserted); | 
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|  | 113 | World::getInstance().signOn(this, World::AtomRemoved); | 
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|  | 114 | AtomObserver::getInstance().signOn(this, AtomObservable::PositionChanged); | 
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|  | 115 | } | 
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|  | 116 |  | 
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|  | 117 | /** Signs off from AtomObserver and World. | 
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|  | 118 | * | 
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|  | 119 | */ | 
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|  | 120 | void LinkedCell::LinkedCell_Model::stopListening() | 
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|  | 121 | { | 
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|  | 122 | World::getInstance().signOff(this, World::AtomInserted); | 
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|  | 123 | World::getInstance().signOff(this, World::AtomRemoved); | 
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|  | 124 | AtomObserver::getInstance().signOff(this, AtomObservable::PositionChanged); | 
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|  | 125 | } | 
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|  | 126 |  | 
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| [d82961] | 127 |  | 
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|  | 128 | /** Allocates as much cells per axis as required by | 
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|  | 129 | * LinkedCell_Model::BoxPartition. | 
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|  | 130 | * | 
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|  | 131 | */ | 
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|  | 132 | void LinkedCell::LinkedCell_Model::AllocateCells() | 
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|  | 133 | { | 
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|  | 134 | // resize array | 
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|  | 135 | tripleIndex index; | 
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|  | 136 | for (int i=0;i<NDIM;i++) | 
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|  | 137 | index[i] = static_cast<LinkedCellArray::index>(Dimensions.at(i,i)); | 
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|  | 138 | N.resize(index); | 
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|  | 139 | ASSERT(getSize(0)*getSize(1)*getSize(2) < MAX_LINKEDCELLNODES, | 
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|  | 140 | "LinkedCell_Model::AllocateCells() - Number linked of linked cell nodes exceded hard-coded limit, use greater edge length!"); | 
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|  | 141 |  | 
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|  | 142 | // allocate LinkedCell instances | 
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|  | 143 | for(index[0] = 0; index[0] != static_cast<LinkedCellArray::index>(Dimensions.at(0,0)); ++index[0]) { | 
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|  | 144 | for(index[1] = 0; index[1] != static_cast<LinkedCellArray::index>(Dimensions.at(1,1)); ++index[1]) { | 
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|  | 145 | for(index[2] = 0; index[2] != static_cast<LinkedCellArray::index>(Dimensions.at(2,2)); ++index[2]) { | 
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|  | 146 | LOG(5, "INFO: Creating cell at " << index[0] << " " << index[1] << " " << index[2] << "."); | 
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|  | 147 | N(index) = new LinkedCell(index); | 
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|  | 148 | } | 
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|  | 149 | } | 
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|  | 150 | } | 
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|  | 151 | } | 
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|  | 152 |  | 
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|  | 153 | /** Frees all Linked Cell instances and sets array dimensions to (0,0,0). | 
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|  | 154 | * | 
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|  | 155 | */ | 
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|  | 156 | void LinkedCell::LinkedCell_Model::Reset() | 
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|  | 157 | { | 
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|  | 158 | // free all LinkedCell instances | 
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|  | 159 | for(iterator3 iter3 = N.begin(); iter3 != N.end(); ++iter3) { | 
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|  | 160 | for(iterator2 iter2 = (*iter3).begin(); iter2 != (*iter3).end(); ++iter2) { | 
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|  | 161 | for(iterator1 iter1 = (*iter2).begin(); iter1 != (*iter2).end(); ++iter1) { | 
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|  | 162 | delete *iter1; | 
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|  | 163 | } | 
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|  | 164 | } | 
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|  | 165 | } | 
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|  | 166 | // set dimensions to zero | 
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|  | 167 | N.resize(boost::extents[0][0][0]); | 
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|  | 168 | } | 
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|  | 169 |  | 
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|  | 170 | /** Inserts all points contained in \a set. | 
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|  | 171 | * | 
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|  | 172 | * @param set set with points to insert into linked cell structure | 
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|  | 173 | */ | 
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|  | 174 | void LinkedCell::LinkedCell_Model::insertPointCloud(IPointCloud &set) | 
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|  | 175 | { | 
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|  | 176 | if (set.IsEmpty()) { | 
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|  | 177 | ELOG(1, "set is NULL or contains no linked cell nodes!"); | 
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|  | 178 | return; | 
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|  | 179 | } | 
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|  | 180 |  | 
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|  | 181 | // put each atom into its respective cell | 
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|  | 182 | set.GoToFirst(); | 
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|  | 183 | while (!set.IsEnd()) { | 
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|  | 184 | TesselPoint *Walker = set.GetPoint(); | 
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|  | 185 | addNode(Walker); | 
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|  | 186 | set.GoToNext(); | 
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|  | 187 | } | 
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|  | 188 | } | 
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|  | 189 |  | 
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|  | 190 | /** Calculates the required edge length for the given desired distance. | 
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|  | 191 | * | 
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|  | 192 | * We need to make some matrix transformations in order to obtain the required | 
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|  | 193 | * edge lengths per axis. Goal is guarantee that whatever the shape of the | 
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|  | 194 | * domain that always all points at least up to \a distance away are contained | 
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|  | 195 | * in the nearest neighboring cells. | 
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|  | 196 | * | 
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|  | 197 | * @param distance distance of this linked cell array | 
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|  | 198 | */ | 
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|  | 199 | void LinkedCell::LinkedCell_Model::setPartition(double distance) | 
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|  | 200 | { | 
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|  | 201 | // generate box matrix of desired edge length | 
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|  | 202 | RealSpaceMatrix neighborhood; | 
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|  | 203 | neighborhood.setIdentity(); | 
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|  | 204 | neighborhood *= distance; | 
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|  | 205 |  | 
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|  | 206 | // obtain refs to both domain matrix transformations | 
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|  | 207 | //const RealSpaceMatrix &M = domain.getM(); | 
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|  | 208 | const RealSpaceMatrix &Minv = domain.getMinv(); | 
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|  | 209 |  | 
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|  | 210 | RealSpaceMatrix output = Minv * neighborhood; | 
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|  | 211 |  | 
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|  | 212 | std::cout << Minv << " * " << neighborhood << " = " << output << std::endl; | 
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|  | 213 |  | 
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|  | 214 | Dimensions = output.invert(); | 
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|  | 215 | Partition = Minv*Dimensions; // | 
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|  | 216 |  | 
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|  | 217 | std::cout << "Dimensions are then " << Dimensions << std::endl; | 
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|  | 218 | std::cout << "Partition matrix is then " << Partition << std::endl; | 
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|  | 219 | } | 
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|  | 220 |  | 
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|  | 221 | /** Returns the number of required neighbor-shells to get all neighboring points | 
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|  | 222 | * in the given \a distance. | 
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|  | 223 | * | 
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|  | 224 | * @param distance radius of neighborhood sphere | 
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|  | 225 | * @return number of LinkedCell's per dimension to get all neighbors | 
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|  | 226 | */ | 
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|  | 227 | const LinkedCell::tripleIndex LinkedCell::LinkedCell_Model::getStep(const double distance) const | 
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|  | 228 | { | 
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|  | 229 | tripleIndex index; | 
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|  | 230 | index[0] = index[1] = index[2] = 0; | 
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|  | 231 |  | 
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|  | 232 | if (fabs(distance) < std::numeric_limits<double>::min()) | 
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|  | 233 | return index; | 
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|  | 234 | // generate box matrix of desired edge length | 
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|  | 235 | RealSpaceMatrix neighborhood; | 
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|  | 236 | neighborhood.setIdentity(); | 
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|  | 237 | neighborhood *= distance; | 
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|  | 238 |  | 
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|  | 239 | const RealSpaceMatrix output = Partition * neighborhood; | 
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|  | 240 |  | 
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|  | 241 | //std::cout << "GetSteps: " << Partition << " * " << neighborhood << " = " << output << std::endl; | 
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|  | 242 |  | 
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|  | 243 | const RealSpaceMatrix steps = output; | 
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|  | 244 | for (size_t i =0; i<NDIM; ++i) | 
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|  | 245 | index[i] = ceil(steps.at(i,i)); | 
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|  | 246 | LOG(2, "INFO: number of shells are ("+toString(index[0])+","+toString(index[1])+","+toString(index[2])+")."); | 
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|  | 247 |  | 
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|  | 248 | return index; | 
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|  | 249 | } | 
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|  | 250 |  | 
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|  | 251 | /** Calculates the index of the cell \a position would belong to. | 
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|  | 252 | * | 
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|  | 253 | * @param position position whose associated cell to calculate | 
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|  | 254 | * @return index of the cell | 
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|  | 255 | */ | 
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|  | 256 | const LinkedCell::tripleIndex LinkedCell::LinkedCell_Model::getIndexToVector(const Vector &position) const | 
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|  | 257 | { | 
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|  | 258 | tripleIndex index; | 
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|  | 259 | Vector x(Partition*position); | 
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|  | 260 | LOG(2, "INFO: Transformed position is " << x << "."); | 
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|  | 261 | for (int i=0;i<NDIM;i++) { | 
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|  | 262 | index[i] = static_cast<LinkedCellArray::index>(floor(x[i])); | 
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|  | 263 | } | 
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|  | 264 | return index; | 
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|  | 265 | } | 
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|  | 266 |  | 
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|  | 267 | /** Adds a node to the linked cell structure | 
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|  | 268 | * | 
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|  | 269 | * @param Walker node to add | 
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|  | 270 | */ | 
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| [029870] | 271 | void LinkedCell::LinkedCell_Model::addNode(const TesselPoint *Walker) | 
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| [d82961] | 272 | { | 
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|  | 273 | tripleIndex index = getIndexToVector(Walker->getPosition()); | 
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|  | 274 | LOG(2, "INFO: " << *Walker << " goes into cell " << index[0] << ", " << index[1] << ", " << index[2] << "."); | 
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|  | 275 | LOG(2, "INFO: Cell's indices are " | 
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|  | 276 | << N(index)->getIndex(0) << " " | 
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|  | 277 | << N(index)->getIndex(1) << " " | 
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|  | 278 | << N(index)->getIndex(2) << "."); | 
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|  | 279 | N(index)->addPoint(Walker); | 
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|  | 280 | std::pair<MapPointToCell::iterator, bool> inserter = CellLookup.insert( std::make_pair(Walker, N(index)) ); | 
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|  | 281 | ASSERT( inserter.second, | 
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|  | 282 | "LinkedCell_Model::addNode() - Walker " | 
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|  | 283 | +toString(*Walker)+" is already present in cell " | 
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|  | 284 | +toString((inserter.first)->second->getIndex(0))+" " | 
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|  | 285 | +toString((inserter.first)->second->getIndex(1))+" " | 
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|  | 286 | +toString((inserter.first)->second->getIndex(2))+"."); | 
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|  | 287 | } | 
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|  | 288 |  | 
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|  | 289 | /** Removes a node to the linked cell structure | 
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|  | 290 | * | 
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|  | 291 | * We do nothing of Walker is not found | 
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|  | 292 | * | 
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|  | 293 | * @param Walker node to remove | 
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|  | 294 | */ | 
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|  | 295 | void LinkedCell::LinkedCell_Model::deleteNode(const TesselPoint *Walker) | 
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|  | 296 | { | 
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|  | 297 | MapPointToCell::iterator iter = CellLookup.begin(); | 
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|  | 298 | for (; iter != CellLookup.end(); ++iter) | 
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|  | 299 | if (iter->first == Walker) | 
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|  | 300 | break; | 
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|  | 301 | ASSERT(iter != CellLookup.end(), | 
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|  | 302 | "LinkedCell_Model::deleteNode() - Walker not present in cell stored under CellLookup."); | 
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|  | 303 | if (iter != CellLookup.end()) { | 
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|  | 304 | CellLookup.erase(iter); | 
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|  | 305 | iter->second->deletePoint(Walker); | 
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|  | 306 | } | 
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|  | 307 | } | 
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|  | 308 |  | 
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|  | 309 | /** Move Walker from current cell to another on position change. | 
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|  | 310 | * | 
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|  | 311 | * @param Walker node who has moved. | 
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|  | 312 | */ | 
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|  | 313 | void LinkedCell::LinkedCell_Model::moveNode(const TesselPoint *Walker) | 
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|  | 314 | { | 
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|  | 315 | ASSERT(0, "LinkedCell_Model::moveNode() - not implemented yet."); | 
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|  | 316 | } | 
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|  | 317 |  | 
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|  | 318 | /** Checks whether cell indicated by \a relative relative to LinkedCell_Model::internal_index | 
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|  | 319 | * is out of bounds. | 
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|  | 320 | * | 
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|  | 321 | * \note We do not check for boundary conditions of LinkedCell_Model::domain, | 
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|  | 322 | * we only look at the array sizes | 
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|  | 323 | * | 
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|  | 324 | * @param relative index relative to LinkedCell_Model::internal_index. | 
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|  | 325 | * @return true - relative index is still inside bounds, false - outside | 
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|  | 326 | */ | 
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|  | 327 | bool LinkedCell::LinkedCell_Model::checkArrayBounds(const tripleIndex &index) const | 
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|  | 328 | { | 
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|  | 329 | bool status = true; | 
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|  | 330 | for (size_t i=0;i<NDIM;++i) { | 
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|  | 331 | status = status && ( | 
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|  | 332 | (index[i] >= 0) && | 
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|  | 333 | (index[i] < getSize(i)) | 
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|  | 334 | ); | 
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|  | 335 | } | 
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|  | 336 | return status; | 
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|  | 337 | } | 
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|  | 338 |  | 
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|  | 339 | /** Corrects \a index according to boundary conditions of LinkedCell_Model::domain . | 
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|  | 340 | * | 
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|  | 341 | * @param index index to correct according to boundary conditions | 
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|  | 342 | */ | 
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|  | 343 | void LinkedCell::LinkedCell_Model::applyBoundaryConditions(tripleIndex &index) const | 
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|  | 344 | { | 
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|  | 345 | for (size_t i=0;i<NDIM;++i) { | 
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|  | 346 | switch (domain.getConditions()[i]) { | 
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|  | 347 | case Box::Wrap: | 
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|  | 348 | if ((index[i] < 0) || (index[i] >= getSize(i))) | 
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|  | 349 | index[i] = (index[i] % getSize(i)); | 
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|  | 350 | break; | 
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|  | 351 | case Box::Bounce: | 
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|  | 352 | break; | 
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|  | 353 | case Box::Ignore: | 
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|  | 354 | break; | 
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|  | 355 | default: | 
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|  | 356 | ASSERT(0, "LinkedCell_Model::checkBounds() - unknown boundary conditions."); | 
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|  | 357 | break; | 
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|  | 358 | } | 
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|  | 359 | } | 
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|  | 360 | } | 
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|  | 361 |  | 
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|  | 362 | /** Calculates the interval bounds of the linked cell grid. | 
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|  | 363 | * \param index index to give relative bounds to | 
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|  | 364 | * \param step how deep to check the neighbouring cells (i.e. number of layers to check) | 
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|  | 365 | * \return pair of tripleIndex indicating lower and upper bounds | 
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|  | 366 | */ | 
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|  | 367 | const LinkedCell::LinkedCell_Model::LinkedCellNeighborhoodBounds LinkedCell::LinkedCell_Model::getNeighborhoodBounds( | 
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|  | 368 | const tripleIndex &index, | 
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|  | 369 | const tripleIndex &step | 
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|  | 370 | ) const | 
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|  | 371 | { | 
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|  | 372 | LinkedCellNeighborhoodBounds neighbors; | 
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|  | 373 |  | 
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|  | 374 | // calculate bounds | 
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|  | 375 | for (size_t i=0;i<NDIM;++i) { | 
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|  | 376 | switch (domain.getConditions()[i]) { | 
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|  | 377 | case Box::Wrap: | 
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|  | 378 | if (index[i] - step[i] >= 0) { | 
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|  | 379 | neighbors.first[i] = index[i] - step[i]; | 
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|  | 380 | } else { | 
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|  | 381 | neighbors.first[i] = getSize(i) - (index[i] - step[i]); | 
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|  | 382 | } | 
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|  | 383 | if (index[i] + step[i] < getSize(i)) { | 
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|  | 384 | neighbors.second[i] = index[i] + step[i]; | 
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|  | 385 | } else { | 
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|  | 386 | neighbors.second[i] = index[i] + step[i] - getSize(i); | 
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|  | 387 | } | 
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|  | 388 | break; | 
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|  | 389 | case Box::Bounce: | 
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|  | 390 | if (index[i] - step[i] >= 0) { | 
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|  | 391 | neighbors.first[i] = index[i] - step[i]; | 
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|  | 392 | } else { | 
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|  | 393 | neighbors.first[i] = - (index[i] - step[i]); | 
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|  | 394 | } | 
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|  | 395 | if (index[i] + step[i] < getSize(i)) { | 
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|  | 396 | neighbors.second[i] = index[i] + step[i]; | 
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|  | 397 | } else { | 
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|  | 398 | const size_t shift = index[i] + step[i]; | 
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|  | 399 | const size_t mod = shift / getSize(i); | 
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|  | 400 | if ((mod / 2)*2 == mod) // even -> come in from lower bound | 
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|  | 401 | neighbors.second[i] = (shift % getSize(i)); | 
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|  | 402 | else // odd -> come in from upper bound | 
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|  | 403 | neighbors.second[i] = getSize(i) - (shift % getSize(i)); | 
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|  | 404 | } | 
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|  | 405 | break; | 
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|  | 406 | case Box::Ignore: | 
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|  | 407 | ASSERT(index[i] - step[i] >= 0, | 
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|  | 408 | "LinkedCell_Model::getNeighborhoodBounds() - lower out of bounds on 'Ignore' boundary conditions"); | 
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|  | 409 | neighbors.first[i] = index[i] - step[i]; | 
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|  | 410 | ASSERT (index[i] + step[i] < getSize(i), | 
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|  | 411 | "LinkedCell_Model::getNeighborhoodBounds() - upper out of bounds on 'Ignore' boundary conditions"); | 
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|  | 412 | neighbors.second[i] = index[i] + step[i]; | 
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|  | 413 | ASSERT(neighbors.second[i] < neighbors.first[i], | 
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|  | 414 | "LinkedCell_Model::getNeighborhoodBounds() - [" | 
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|  | 415 | +toString(neighbors.first[i])+","+toString(neighbors.second[i]) | 
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|  | 416 | +"] does not make sense as array bounds."); | 
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|  | 417 | break; | 
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|  | 418 | default: | 
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|  | 419 | ASSERT(0, "LinkedCell_Model::getNeighborhoodBounds() - unknown boundary conditions."); | 
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|  | 420 | break; | 
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|  | 421 | } | 
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|  | 422 | } | 
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|  | 423 |  | 
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|  | 424 | return neighbors; | 
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|  | 425 | } | 
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| [2614e2a] | 426 |  | 
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|  | 427 | /** Callback function for Observer mechanism. | 
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|  | 428 | * | 
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|  | 429 | * @param publisher reference to the Observable that calls | 
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|  | 430 | */ | 
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|  | 431 | void LinkedCell::LinkedCell_Model::update(Observable *publisher) | 
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|  | 432 | {} | 
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|  | 433 |  | 
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|  | 434 | /** Callback function for the Notifications mechanism. | 
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|  | 435 | * | 
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|  | 436 | * @param publisher reference to the Observable that calls | 
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|  | 437 | * @param notification specific notification as cause of the call | 
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|  | 438 | */ | 
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|  | 439 | void LinkedCell::LinkedCell_Model::recieveNotification(Observable *publisher, Notification_ptr notification) | 
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|  | 440 | {} | 
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|  | 441 |  | 
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|  | 442 | /** Callback function when an Observer dies. | 
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|  | 443 | * | 
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|  | 444 | * @param publisher reference to the Observable that calls | 
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|  | 445 | */ | 
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|  | 446 | void LinkedCell::LinkedCell_Model::subjectKilled(Observable *publisher) | 
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|  | 447 | {} | 
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|  | 448 |  | 
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