Understanding Cell Nets

When Cells are linked together via coupled mass flux links, GoldSim treats the network of Cells as a single coupled system of differential equations, solving for the mass of each species The chemical (or non-chemical, such as bacterial or viral) constituents that are stored and transported through an environmental system in a contaminant transport model. In GoldSim, the Species element defines all of the contaminant species being simulated (and their properties). in each medium in each Cell as a function of time.

In order to solve these equations, GoldSim initially identifies the sub-networks of interconnected Cells, referred to as cell nets, which then are treated as single entities in the calculations. GoldSim sets up and solves a separate set of fully coupled equations for each net. Hence, the fluxes between cells within a cell net A network of interconnected Cell pathways that are directly linked by coupled mass flux links. are computed simultaneously (since they are solved as a system of coupled differential equations). Fluxes between cell nets (or between cell nets and non-Cell pathways), however, are propagated asynchronously (i.e., the "upstream" pathway or cell net is solved first, followed by the "downstream" one).

A cell net is defined by all Cells interconnected via coupled mass flux links. Note that since only Cell to Cell mass flux links can be coupled, two Cells which are linked via mass flux links but are separated by a non-Cell pathway A transport pathway element that is mathematically equivalent to a finite difference node. Cells are commonly applied to simulate discrete compartments in an environmental system (such as ponds, lakes, shallow soil compartments, or the atmosphere). (e.g., a Pipe), are, by definition, in different cell nets. Also, if two Cells are connected by normal mass flux links (rather than coupled links), they will be in different cell nets.

Note: Cell nets cannot cross between SubSystems. A SubSystem A specialized Container that is completely “self-contained”. SubSystems can take on some useful features and properties (e.g., conditionality, having an internal clock, and being able to loop), but also have some limitations (with regard to how they can be incorporated into feedback loops). is a specialized Container An element that acts like a "box" or a "folder" into which other elements can be placed. It can be used to create hierarchical models. in GoldSim that is entirely self-contained. Hence, by definition, Cell nets cannot extend across a SubSystem boundary.

The example below shows a pathway network in a Container, illustrating schematically how GoldSim would define the cell nets.

In general terms, a transport pathway Physical components or compartments through which contaminant species can move and/or be stored, such as aquifers, lakes, sediments, surface soil and the atmosphere. GoldSim provides four different elements for simulating pathways. can be thought of as a function whose input is a mass rate, and whose output is a mass rate. This statement is not strictly true for all types of pathways. In particular, the statement is true for non-Cell pathways, but is not strictly true for Cell pathways (since mass transfer to or from a Cell with diffusive connections would not necessarily be unidirectional). It is, however, true for cell nets. Hence, each cell net can be thought of as a very complex sort of pathway, which acts as a transfer function having multiple unidirectional input and output connections.

The equations GoldSim solves for each net are defined in terms of nodes, where a node represents either a Cell in the cell net (primary nodes) or monitored discharges from the net (secondary nodes). GoldSim defines a secondary node for any mass flux link An interconnnection between two transport pathways that defines the rate at which species move between the pathways. for which it needs to compute a flux rate:

• mass flux links to non-Cell pathways or to other cell nets;
• mass flux links within a cell net for which the user has requested that results be saved; and
• mass flux links which are referenced in other elements (e.g., Sum or Extrema) for which the user has requested that results be saved.

GoldSim solves a matrix A two-dimensional array. equation representing the coupled system of equations based on the backwards-difference (fully implicit) algorithm for each cell net and each species decay-chain family to compute concentrations and fluxes. GoldSim uses an iterative, sparse-matrix approach (the bi-conjugate gradient stabilized method) to solve the system of equations.

The backwards difference (fully implicit) algorithm has the advantages of being unconditionally stable, conserving mass, and having bounded error levels. In order to enhance the accuracy of the backwards-difference method, GoldSim automatically divides every time step into a number of fractional timesteps, depending on the user's selected precision level (1 for Low, 4 for Medium and 10 for High). Other solution parameters (e.g., the overall solver error tolerance, as well as a minimum number of solver iterations, how solubility constraints are represented) are also determined by the precision setting.

Using fractional timesteps is a very computationally efficient way to enhance the accuracy of the solution. This is because 1) only the cell nets are updated during these fractional timesteps; and 2) numerically the fractional timesteps are computed via matrix calculations in such a way that it takes less computational effort than simply shortening the actual timestep A discrete interval of time used in dynamic simulations.. However, fractional timesteps can also have a detrimental affect (i.e., produce inaccuracies) when the media Materials (such as water, sand, clay, air) that constitute (are contained within) transport pathways. GoldSim provides two types of elements for defining media: Fluids and Solids. amounts (e.g., the volume of water) is rapidly changing in a Cell. In these cases, GoldSim will write warning messages to the Run Log Text that is stored with a GoldSim model once it has been run. It contains basic information regarding the simulation, and any warning or error messages that were generated., and there are actions you can take to minimize any errors.

The computational effort to solve the coupled equations of a cell net increases rapidly (i.e., not linearly) with the size of the cell net. In almost all models, this is not a problem, but for extremely large models (hundreds of Cells), it may be worthwhile to break a very large cell net into several smaller cell nets (by using a Normal link A standard link between two elements. in place of a Couple link when connecting Cells).

Note that because cell nets are computationally treated as a single entity, they are presented as such in the Function of and Affects view: