Abstract: One embodiment of the present invention sets forth a technique for efficiently simulating breaking waves in real-time. A two-dimensional shallow water height field simulation generates height and velocity information used to generate a wave line for each wave within the height field that satisfies criteria for overturning. For each overturning wave, a wave sheet is created from particles generated relative to points on the respective wave line. Each wave sheet may move separately from an underlying wave that gave rise to the wave sheet, allowing the wave sheet to fall and break, creating a realistic appearance. As a falling wave sheet collides with the underlying wave or water surface, free particles may be generated to simulation spray visible on a real breaking wave.

Abstract: A model of at least one rheological characteristic of a fluid is provided. According to the model, a behavior of the tool that includes an actuator that uses the fluid is predicted. The behavior of the tool is predicted for at least one expected condition in a well.

Abstract: Embodiments include a system for determining patient-specific cardiovascular information. The system may include at least one computer system configured to receive patient-specific data regarding a geometry of an anatomical structure of the patient and create a three-dimensional model representing at least a portion of the anatomical structure of the patient based on the patient-specific data. The at least one computer system may be further configured to determine a total resistance associated with a total flow through the portion of the anatomical structure of the patient, and determine information regarding a blood flow characteristic within the anatomical structure of the patient based on the three-dimensional model, a physics-based model relating to the anatomical structure of the patient, and the determined total resistance.

Abstract: A medical practitioner can specify certain parameters for a procedure that involves delivering a therapeutic agent, while leaving other parameters open. The therapeutic agent can be sensitive to biomechanical forces (or other influences) associated with delivery. The procedure can involve regenerative medicine, for example delivering progenitor or stem cells to a diseased heart using a catheter, whereby unbridled transport in the catheter may compromise efficacy. The open parameters can influence efficacy of the agent and thus therapeutic outcome. A computer-based system can apply stored information, such as from databases, to narrow the possible values of the open parameters. From the narrowed possibilities, an optimization routine can determine suitable or optimized values for the open parameters. The determined values can manage biomechanical forces incurred by the therapeutic agent, thereby promoting efficacy and healing. The optimized parameters can guide the practitioner in the procedure.

Abstract: Computational methods are used to create cardiovascular simulations having desired hemodynamic features. Cardiovascular modeling methods produce descriptions of blood flow and pressure in the heart and vascular networks. Numerical methods optimize and solve nonlinear equations to find parameter values that result in desired hemodynamic characteristics including related flow and pressure at various locations in the cardiovascular system, movements of soft tissues, and changes for different physiological states. The modeling methods employ simplified models to approximate the behavior of more complex models with the goal of to reducing computational expense. The user describes the desired features of the final cardiovascular simulation and provides minimal input, and the system automates the search for the final patient-specific cardiovascular model.

Abstract: Systems and methods of simulating an explosion in time-marching finite element analysis are disclosed in the present invention. A method is configured for increasing user (e.g., engineer or scientist) productivity by reducing computation time of simulating fluid-structure interaction due to an explosion. The method comprises a creation of a finite element analysis model that includes structure, surrounding fluid, a blast source of the explosion and a single layer of ambient elements each having a segment representing a boundary of the fluid facing the blast source. Each ambient element is associated with a particular finite element representing the fluid at the boundary. The ambient elements are configured to be situated between the blast source and the structure such that the simulation can be carried on a set of boundary conditions specified thereon. The boundary conditions comprise a set of nodal velocities that are determined from the empirical formula (e.g., Friedlander equation).

Abstract: A method of generating a mesh of an object (11) that moves through a fluid medium which is used in the design or analysis of the object (11) in connection with a hybrid method combining RANS and LES comprising the following steps: a) creating an inner “C” topology (21) around the object (11) for a boundary layer description; b) creating an outer “C” topology (23) covering a space region including the separation region; c) locating the separation region and adapting to it the inner and outer “C's” topologies (21, 23) so that the outer “C” topology (23) is adapted to the size of the separated region and the inner “C” topology (21) is adapted to the size of the boundary layer; d) refining the mesh in the separation region according to the specifications of the RANS/LES method. The invention also provides a system for carrying out the method.

Abstract: A pressure regulator for controlling pressure with high response and high accuracy even a flow change occurs on the downstream side. The pressure regulator (1) regulates the rate of inflow into an isothermal pressure vessel (13) for the gas supplied from a gas supply source (10) by means of a servo valve (11) to maintain the constant pressure in the isothermal pressure vessel (13).

Abstract: A database design modeling system for managing data facets for design scenarios as a collection of hierarchical unitized data alternatives is provided. The database design model allows for easy access to data while providing a single persistant location for any one specific data record, thereby reducing any problems associated with conventional data duplication. Further, all data stored within the database is unitized thus eliminating the need for the user to do the unit mapping.

Abstract: Method for making a probabilistic determination of total seal capacity for a hydrocarbon trap, simultaneously considering both capillary entry pressure and mechanical seal capacity, and where capillary entry pressure is estimated by relating it directly to the buoyancy pressure applied by the hydrocarbon column to the top seal. The method thus considers the substantial uncertainty associated with input parameters, which uncertainty limits the utility of such analyses for robust hydrocarbon column height and fluid contact predictions. The method disclosed for estimating seal capillary entry pressure, the requisite input parameter for capillary seal capacity analysis, by inverting trap parameters avoids the need for direct measurement by mercury injection capillary capacity tests on small pieces of rock, which test results often are not available for all desired locations nor are they necessarily representative of adjacent rocks in the seal.

Abstract: A computer software program provides an algorithm that solves for unknown demands (and junction pressures) within a modeling system that uses a generalized, unified loop-node formulation. The program can be used to calculate the available demand (i.e., the amount of water that is to be supplied) according to the nodal pressure. Both nodal heads and flows are simultaneously solved using a gradient algorithm, which allows, in accordance with the present invention, the model to simulate situations where a change in pressure affects the quantity of water used. Criticality analyses for segments of a system in such pressure dependent scenarios can also be performed using the software program of the present invention.

Abstract: System and method for extraction and processing of river bank coordinates from imagery, generation of an unstructured mesh of the river using river bank positions and available or synthetic bathymetry, application of upstream and downstream boundary forcing data, contingencies for handling missing data, and configuration of multiple realizations of the developed river model.

Abstract: A method of water erosion management comprises receiving a topographic datum indicating the topography of a defined area. Data is received indicating a soil composition found at the defined area. A model is created predicting a water runoff pattern for said defined area based upon the topography of the defined area and the soil composition of the defined area.

Abstract: A method for estimating properties of fluids in rock formations at selected locations within a geologic basin includes generating an initial model of the basin. The model includes as output spatial distribution of a rock formation mineral composition, rock formation porosity and composition of fluids in the rock formation porosity. An initial estimate of fluid composition is generated for a rock formation, and a sample of fluid is extracted from pore spaces of the rock formation. Composition of the fluid sample is analyzed from within, and the initial model of the basin is adjusted to substantially match the analyzed fluid composition. The adjusted model is used to generate an initial estimate of fluid composition.

Abstract: A method, apparatus and computer-readable medium for estimating sand production from an earth formation is disclosed. A grain-scale formation model of the earth formation is created, wherein a value obtained from the grain-scale formation model compares to a value of a measured property of the earth formation. A fluid parameter of the grain-scale formation model is determined and a movement of at least one grain of the grain-scale formation model is determined due to the determined fluid parameter. The sand production is estimated from the determined movement of the at least one grain.

Abstract: Simulation of thermal fluid-structure interaction using bulk flow fluid elements (BFFEs) is described. Each BFFE is configured to include the following characteristics: 1) at least one surrounding layer of solid elements representing either the surrounding structure or the pipe wall; 2) a layer of shell elements or Bulk Node Segments representing the outer boundary of the fluid; 3) a Bulk Node at the center of the BFFE for defining fluid properties (e.g., density, specific heat) and volume (i.e., fluid volume is calculated as the enclosed volume between the Bulk Node and all of the Bulk Node Segments that surround it); 4) a fluid flow beam element or Bulk Node Element for defining fluid flow path to another BFFE; and 5) a contact interface between the solid elements and the shell elements for conducting fluid-structure thermal interaction.

Abstract: A core column simulator device and methods of assessing the efficacy of a wastewater biotreatment system using the core column simulator device are disclosed.

Abstract: Methods, systems, and computer program products are provided for modeling movement of air underneath a floor of a data center. Embodiments include receiving, by a management system, from a plurality of environmental condition sensors, environmental condition information corresponding to an area underneath the floor of the data center; in dependence upon the environmental condition information, generating, by the management system, a model representing movement of air within the area underneath the floor of the data center; and indicating, by the management system, physical obstacles underneath the floor of the data center based on the model of the movement of air.

Abstract: A method and system for non-invasive patient-specific assessment of coronary artery disease is disclosed. An anatomical model of a coronary artery is generated from medical image data. A velocity of blood in the coronary artery is estimated based on a spatio-temporal representation of contrast agent propagation in the medical image data. Blood flow is simulated in the anatomical model of the coronary artery using a computational fluid dynamics (CFD) simulation using the estimated velocity of the blood in the coronary artery as a boundary condition.

Abstract: Diagnostic and response systems and methods for a fluid power system acquire data from pressure and temperature sensors disposed in the fluid power system, analyze the data in a failure algorithm to build a history of cumulative damage to hoses in the fluid power system, communicates an indication of potential imminent hose failure to a central location when a level of the cumulative damage indicates imminent failure of a hose, analyze the information at the central location to determine an appropriate response, and transmit information about the fluid power system, including location, and identification of the hose about to fail to a response unit. The response unit responds to the location and replaces the component prior to failure, or the communication might include information that the hose has failed, such that the response unit replaces the failed hose to return the fluid power system to normal operation.

Abstract: To update particle information in a particulate fluid flow simulation, the new positions of the particles are determined by the balance of all forces, and such forces are calculated by taking surface integrals along the particle-fluid interface to reduce CPU overhead. A 3-D collision scheme for ellipsoidal particles to determine whether two or more ellipsoidal particles in the fluid flow collide, and if so, determine the location of the collision point between two colliding particles is also disclosed. Other aspects involve predicting the new location and orientation of each of the colliding particles after collision.

Abstract: A method of predicting the occurrence of a wind gust at a wind turbine is described. The values of at least one parameter measured for the wind turbine as input data are obtained on a predetermined frequency. A group of input data of predetermined size that was most recently obtained is selected. The probability of the occurrence of a wind gust is calculated based on the selected group of input data. The calculated probability that a wind gust will occur is verified to be above a predetermined probability level. And the occurrence of the wind gust is predicted.

Abstract: A method for determining resistivity distribution of formations below a bottom of a body of water from transient electromagnetic signals acquired by imparting a transient electromagnetic field into the water and detecting an electromagnetic response thereto at a plurality of spaced apart positions from a place of the imparting includes simulating an air wave response at each of the plurality of spaced apart positions. The simulated air wave response is subtracted from the detected response to produce a subsurface impulse response at each of the plurality of positions. The subsurface impulse responses are used to determine the resistivity distribution.

Abstract: A simple and efficient method for simulating dispersed bubble flow is provided. Instead of modeling the complex hydrodynamics of numerous small bubbles explicitly, the method approximates the average motion of these bubbles using a continuum multiphase solver. The subgrid interactions among bubbles are computed using a new stochastic solver. Using the proposed scheme, complex scenes with millions of bubbles can be simulated efficiently.

Abstract: Embodiments include a system for providing patient-specific cardiovascular information using a web site. The system may include at least one computer system configured to allow a remote user to access a web site, receive patient-specific data regarding at least a portion of a geometry of an anatomical structure of the patient, and create a three-dimensional model representing at least a portion of the anatomical structure of the patient based on the patient-specific data. The at least one computer system may be further configured to determine information regarding a blood flow characteristic within the anatomical structure of the patient based on the three-dimensional model and a physiological condition of the patient, and communicate display information regarding a first three-dimensional simulation of at least the portion of the anatomical structure of the patient to the remote user using the web site. The three-dimensional simulation may include the determined information.

Abstract: A flood modeling system defines virtual cells of a geographic region. Some of the cells, referred to as “channel cells,” represent areas within a channel for a moving body of water, such as a stream or river. Other cells, referred to as “land cells,” represent areas external to the channels within the geographic region. The flow of water through the geographic region is modeled as a virtual flow of water through the channel cells and the land cells. The virtual flow of water through the channel cells is modeled via a channel flow algorithm, and the virtual flow of water through the land cells is modeled via a land flow algorithm. For each time step, the system calculates the amount of water that virtually flows out of one cell into adjacent cells. Thus, for any given time step, each cell has an estimated amount of water virtually contained within the cell. The water estimated to virtually flow above ground represents flood water and can be quantified for each cell.

Abstract: Embodiments include a system for planning treatment for a patient. The system may include at least one computer system configured to receive patient-specific data regarding a geometry of an anatomical structure of the patient, create a three-dimensional model representing at least a portion of the anatomical structure of the patient based on the patient-specific data, and determine a first fractional flow reserve within the anatomical structure of the patient based on the three-dimensional model and a physics-based model relating to the anatomical structure of the patient. The at least one computer system may be further configured to modify the three-dimensional model, and determine a second fractional flow reserve within the anatomical structure of the patient based on the modified three-dimensional model.

Abstract: A method for building robust model predictive controller universally applicable is presented based on the innate process characteristics independent of the method of control actuation. The method of universal MPC design permits proper configuration of requisite regulatory control loops for measured and unmeasured disturbance rejections consistent with the underlying innate process characteristics and their embedding within the overall process unit model predictive controller. The method of universal MPC design requires that manipulated variables process value based model (PV-based models) be used in control and optimization in place of the customary set point based models (SP-based models) or control output based models (OP-based models). The PV-based models are devoid of the manipulated variables regulatory controllers response and tuning. Based on the PV-based models, an alternate method of MPC called PV-based MPC is presented that is most robust and adaptable of possible three types of MPC.

Abstract: Blood flow turbulence or stationary blood can occur in vessels of the body, in particular following surgical interventions or minimally invasive interventional procedures, and this can lead to the formation of thrombi. A model of such vessels is obtained on the basis of a 3D image dataset, a simulation is then performed and a check carried out to determine whether turbulence or stationary blood can be demonstrated in the simulation. The model is then modified, successively where necessary, until the simulation reveals no more turbulence. Then, on the basis of the most recently modified model that results in no turbulence and no stationary blood, an aid is provided, in particular a specific aid such as e.g. a stent produced or a screen display presented.

Abstract: Embodiments include a system for determining cardiovascular information for a patient. The system may include at least one computer system configured to receive patient-specific data regarding a geometry of the patient's heart, and create a three-dimensional model representing at least a portion of the patient's heart based on the patient-specific data. The at least one computer system may be further configured to create a physics-based model relating to a blood flow characteristic of the patient's heart and determine a fractional flow reserve within the patient's heart based on the three-dimensional model and the physics-based model.

Abstract: The invention relates to a method for estimating heat transfer during water quench of an aluminum part. The method includes: estimating the heat transfer of the aluminum part when a temperature of the part is greater than 500° C. using q=?(?T) ??(1); estimating the heat transfer of the aluminum part when the temperature of the part is greater than T2 and less than 500° C.

Abstract: A method for simulating a blood flow in a vascular segment of a patient is proposed. A 3D image dataset of an examination region is recorded by a radiographic diagnostic device for generating a 3D vascular model. Contrast agent propagation in the examination region is captured by a dynamic 2D angiography method for generating a real 2D angiography recording. A CFD simulation of the blood flow is performed in the 3D vascular model based on a blood flow parameter for generating a virtual 2D angiography recording. A degree of correspondence between the real and the virtual 2D angiography recordings is determined from identical angulation and adjusted recording geometry of the patient and compared with predefinable tolerance values. The CFD simulation is iteratively optimized while changing the blood flow parameter as a function of the comparison. The degree of correspondence is outputted when the optimum CFD simulation is achieved.

Abstract: A system and method for rendering flows and volumes includes the steps of generating a preintegrated table, the preintegrated table having a plurality of entries, each of the plurality of entries having color information and opacity information, generating illustrative effects based on the plurality of entries of the preintegrated table and displaying the illustrative effects.

Abstract: A method is disclosed for simulating the formation of sedimentary deposits. In one embodiment, this method involves, (a) solving a two-dimensional time-dependent map view system of equations for at least flow momentum, flow height, suspended sediment concentration, and entrainment of overlying water, (b) calculating net sediment deposition at each map view location using the flow properties, (c) recording the time-variability of the net sediment deposition.

Abstract: Systems, methods, and program product to synthesize a grass-roots heat exchanger network for non-thermodynamically constrained problems for a plurality of hot process streams to be cooled and a plurality of cold process streams to be heated according to a plurality of utilities targets, are provided. An exemplary system includes a heat exchange network synthesizing computer and heat exchange network synthesizing program product configured to analyze the waste heat recovery problem without decomposition to produce a result having an optimal number of network heat exchanger units through application of advanced process-to-process matching schemes for non-thermodynamically constrained problems.

Abstract: A method of constructing a piecewise linear engine performance model, comprises the steps of: (a) providing a non-linear engine performance model which iteratively calculates engine performance variables y from a plurality of sub-models, each sub-model representing the performance of a component of the engine; (b) defining a plurality of engine power conditions, each engine power condition being defined by one or more engine state variables x; (c) for each of the engine power conditions, using the non-linear engine performance model to determine partial derivatives ?y of the engine performance variables y with respect to engine control variables u and with respect to the engine state variables x, the values of the partial derivatives ?y being determined so that, on integrating the partial derivatives ?y, the error between steady state values for the engine performance variables y calculated by the non-linear engine performance model and steady state values for the engine performance variables y calculated

Abstract: Method of determining the compositional evolution of fluids present in a porous medium as a result of biodegradation. A biodegradation compositional scheme is defined from twelve chemical classes allowing the hydrocarbons and the gases resulting from biodegradation to be described. The initial amount of each one of these chemical classes contained in the fluids before biodegradation is determined. A reaction scheme is then defined for biodegradation, wherein the chemical classes react with electron acceptors according to parallel reactions having different velocities, and each class reacts with these acceptors according to sequential reactions. The biodegradation reaction velocities are determined for each class. The composition of the fluids is then deduced by assessing the amount of each chemical class by applying the reaction scheme. The composition of the biodegraded hydrocarbons, as well as the amount of acid gas and of methane produced, are notably deduced.

Abstract: An information processing apparatus for generating a thermal analysis model for a plurality of component models includes a gap simplification unit configured to simplify a gap area having a gap between the plurality of component models by generating a gap model based on an extracted portion of the gap area and by merging the gap using a modification of a component model which contacts the gap, an extraction unit configured to extract a merged face using the gap model and the merged gap, a calculation unit configured to calculate thermal resistance of the merged face based on a thermal conductivity and the merged face, and an assigning unit configured to assign the thermal resistance to the merged face.

Abstract: A system and method is provided for controlling the operation of notifying drivers of the fluid level. Primary and secondary keys are adapted to be associated to primary and secondary drivers. A key ignition device is positioned on the primary and secondary keys and generate driver status signals. A fluid level sensing device is operable to generate a fluid level signal indicative of the fluid level. A controller is coupled to the fluid level sensing device and the key ignition device. The controller determines whether the driver of the vehicle is the primary or secondary driver in response to the driver status signals. The controller notifies the primary or secondary driver of the fluid level in response to the fluid level signal. The controller selectively controls the operation of notifying the primary or the secondary driver based on whether the driver is the primary or secondary driver.

Abstract: A method and system for patient-specific modeling of the whole heart anatomy, dynamics, hemodynamics, and fluid structure interaction from 4D medical image data is disclosed. The anatomy and dynamics of the heart are determined by estimating patient-specific parameters of a physiological model of the heart from the 4D medical image data for a patient. The patient-specific anatomy and dynamics are used as input to a 3D Navier-Stokes solver that derives realistic hemodynamics, constrained by the local anatomy, along the entire heart cycle. Fluid structure interactions are determined iteratively over the heart cycle by simulating the blood flow at a given time step and calculating the deformation of the heart structure based on the simulated blood flow, such that the deformation of the heart structure is used in the simulation of the blood flow at the next time step.

Abstract: A method for predicting fluid flow is provided. An exemplary embodiment of the method comprises obtaining data describing a stratigraphy, the stratigraphy comprising a fault, predicting a volume of shale in the fault and calculating a thickness of the fault. The exemplary method may also comprise calculating a permeability of the fault based on the volume of shale, the thickness, an estimated shale smear continuity factor and an estimated cataclastic reduction factor. The exemplary method may additionally comprise predicting fluid flow in a subsurface region based on the fault permeability.

Abstract: A tool for assisting in the planning or performing of electrical neuromodulation of a patient's spinal cord. The tool may have various functions and capabilities, including calculating a volume of activation, registering an electrode(s) shown in a radiologic image, constructing functional images of the patient's spinal anatomy, targeting of neuromodulation, finding a functional midline between multiple electrodes, determining the three-dimensional position of multiple electrodes, and/or accommodating for electrode migration. In certain embodiments, the tool can be embodied as computer software or a computer system.

Abstract: Methods and apparatus for fast computation of fluid velocity fields in the presence of objects causing flow discontinuity are disclosed. Rigorous evaluation of perturbed velocity fields, each resulting from placing an object in a bounded space sustaining a reference velocity field, are determined for different orientations of the object with respect to a reference velocity field and different values of the magnitude of the velocity field. A velocity field is determined at selected sampling points within an enclosure surrounding the object. Pre-computed perturbed velocity fields determined for each object of a set of reference objects are stored in a computer memory. Fast real-time computation of velocity-field perturbations caused by presence of objects at arbitrary positions in a space sustaining velocity fields of different magnitudes and directions are realized using the stored pre-computed data.

Abstract: The present invention incorporates a device for measuring fluid movement conditions at a distance, such as LIDAR, into an underlying device for extracting useful work from a moving fluid, such as a wind turbine, for the purpose reducing the cost of designing and manufacturing the underlying device.

Abstract: A method of forming a geologic model of a subsurface region is disclosed. Data related to the subsurface region is obtained. A framework is constructed to represent the subsurface region. A template is selected from a plurality of templates. The selected template provides at least one property that is characteristic of the subsurface region. The selected template is inserted into the framework, to form the geologic model. The geologic model is then outputted.

Abstract: A model creation portion having a obstacle in a space includes two creation portions, one is to separate them and another is to combine them which handles, as porous media, an obstacle in which narrow spaces are regularly arranged. A thermosets flow analysis portion includes two analysis portions to separate and combine them, each of them has a viscosity equation for thermosets. The space/obstacle separation analysis portion analyzes by combining the viscosity equation with conservation equations of mass, momentum and energy, and the space/obstacle combination analysis portion analyzes by combining the viscosity equation with a conservation equation directed to a shape simplified as porous media. Resin flow behavior is analyzed while data in an interface between two models is delivered to each other, to accurately predict filling behavior at the time of injecting resin of thermosets molded articles having the complex obstacles.

Abstract: The preferred embodiment of the instant invention preferably takes place in three stages. In the first stage, a determination is made as to which sands (or other reservoir bodies) are in fluid communication with the existing wells, and the disconnected sands are removed from the fine scale geological model. In the second stage, the transmissibilities in the fine scale model are upscaled to the coarse grid for direct use in the simulation. In the third stage the connected sands only are upscaled and a simulation is carried forward to the point when a new well is to be drilled. The pressure and saturation data are exported to combine with newly connected sands (reservoir bodies), and new pressure and saturation is calculated. The new values are exported in the simulation model and simulation run is continued.

Abstract: The present invention relates to an interventional procedure simulation system and method, comprising a control unit and an interface unit, said control unit communicating with said interface unit to simulate handling of at least one instrument interfaced by said interface unit. The control unit comprises a database of vessels having hierarchy structure, each vessel having a diameter and a stiffness, and said instrument being a tool expendable in a simulated vessel. When the tool is expanded, a geometry of said vessel changes resulting in a fluid flow change.

Abstract: A method and computer-readable medium for modeling smoke propagation in a structure are presented, along with applications of the method. A model is provided of a multiplicity of compartments within the structure, and the presence of a fire within a compartment is determined. The energy and smoke released by the fire is estimated, and smoke flows out of the compartment are determined. Each compartment having a smoke source is analyzed, individually, for the propagation of smoke. Each compartment is assumed to have a homogenous upper smoke layer and a lower cool air layer. The analysis includes a mass and energy balance to determine the change in characteristics of the upper smoke layer, where a known pressure for the lower cool air layer is used for the balance. For multiple smoke sources into a compartment, the contributions to the mass and energy balance of each smoke source are considered additive.

Abstract: Method is provided for simulating a physical process such as fluid flow in porous media by performing a fine-grid calculation of the process in a medium and re-using the fine grid solution in subsequent coarse-grid calculations. For fluid flow in subsurface formations, the method may be applied to optimize upscaled calculation grids formed from geologic models. The method decreases the cost of optimizing a grid to simulate a physical process that is mathematically described by the diffusion equation.