Curve matching for parameter identification

Methods and systems for matching a computed curve to a target curve to enable realistic engineering simulations are disclosed. Discrepancies between computed curve and the target curve are measured, and based on the discrepancies, decisions on how to adjust parametric inputs can be made to achieve an optimal result of simulation. Optimization of parameter identification is achieved by adjusting the parametric inputs of a simulation model such that the discrepancy between the two curves is minimized. Because the points on the two curves to be matched are paired, matching of any two open curves, including hysteretic curves, can be handled. Curves that are complete set apart in their original coordinates can be merged to a common coordinate system for parameter identification without the computational instability problems.

FIELD OF THE INVENTION

The present invention generally relates to system identification used in engineering simulation of measured results, more particularly to methods and systems for matching a computed curve to a target curve to enable the calibration of systems or material properties, so that a realistic engineering simulation of the systems or materials can be carried out in a computer system.

BACKGROUND OF THE INVENTION

In a multiple-input dynamic model, the output is a combination of the response of each of the inputs. If the system inputs are observable, meaning the process or contribution of each input can be readily known or determined, then the output may be controlled by directly changing the related inputs. However, in a large complex system, not every input is observable. For a system with unobservable inputs, because the output cannot be directly related to an input, control and optimization of the system is more difficult. A common approach to determine how each input affects the results of an output is to measure the response of a system with known multiple inputs and try to deduce a mathematical relationship between the inputs and the outputs without going into the details about how the system reacts to each individual input. This kind of approach in determining the input-output relationship is called system identification, or parameter identification.

It is noted that “input” can also be “design”. Each different “input” can be a different “design”.

The goal of performing a system identification process is to produce a result that is as close to the desired physical system output as possible by adjusting the inputs of the system. In numerical simulation applications, the parameter identification, or system identification, is to control the parametric inputs of a mathematical model, which is constructed to simulate a system, such that the output of the simulation is closely matching the system output.

In certain applications, the outputs are presented in two-dimensional (2-D) curves, for example, material properties in terms of strain-stress relationship. Generally, it is desirous for a user (e.g., engineer and/or scientist) to use a computer to generate a computed or simulated curve to match a target curve (e.g., obtained in physical specimen test), such that a computer simulation can be performed to product realistic predictions.

Matching a target curve having characteristic of monotonic increase in one coordinate is very simple, for example, using the ordinate difference between two curves as a system identification parameter.

However, matching a computed curve to a non-monotonically increased target curve is not straightforward, for example, when the target curve exhibits hysteretic behaviors. In this situation, both the target curve and the computed curve have more than one possible ordinate value for each value of the abscissa. Hence, if an abscissa value is used to find the matching value on the computed curve, the interpolated ordinate value would be non-unique.

Another problem in matching two curves is related to engineering optimization. During an iterating interim stage of optimization, the computed curve and the target curves can be completely disengaged, i.e., they do not share any common value in one coordinate of a 2-D curve. Under this situation, the optimization iteration process may become computational unstable and fails.

Therefore, it would be desirable to have an improved method and system for matching a computed curve to a target curve to enable realistic engineering simulations.

SUMMARY OF THE INVENTION

This section is for the purpose of summarizing some aspects of the present invention and to briefly introduce some preferred embodiments. Simplifications or omissions in this section as well as in the abstract and the title herein may be made to avoid obscuring the purpose of the section. Such simplifications or omissions are not intended to limit the scope of the present invention.

The present invention pertains to methods and systems for matching a computed curve to a target curve to enable realistic engineering simulations, for example, material behavior. Discrepancies between the computed curve and the target curve are measured, and based on the discrepancies, decisions on how to adjust parametric inputs can be made to achieve an optimal result of simulation. An example of a target curve in engineering simulation is a stress-strain relationship of a material obtained in a specimen test (e.g., stress versus stretch-ratio curve for polymeric material shown inFIG. 1).

Optimization of parameter identification is achieved by adjusting the parametric inputs of a simulation model such that the discrepancy between the two curves is minimized. Because the points on the two curves to be matched are paired, matching of any two open curves, including hysteretic curves, can be handled. Curves that are complete set apart in their original coordinates can be merged to a common coordinate system for parameter identification without the computational instability problems.

According to one aspect of the present invention, a target curve is represented by “m” points (a first set of points) while a computed curve represented by “n” points (a second set of points). The computed curve is created from a computer aided analysis (CAE) model (e.g., finite element analysis model) having a plurality of adjustable control or design parameters.

Coordinates of the first set and second set of points are normalized to their respective smallest bounding ranges. The segment and total lengths of each curve are calculated using the normalized coordinates. A segment length is a straight line distance between two adjacent points. The total length is the sum of all segments in each curve. The ratio between each segment length and the total length is referred to as a length ratio, which is used for creating a third set of points placed on the computed curve. Each of the third set of points is associated with corresponding one of the first set of points based on respective length ratios of the target curve. In other words, the length ratios as result of the third set of points for the computed curve matches the length ratios of the first set of points for the target curve. A discrepancy measurement is carried out between the computed curve and the target curve is then conducted in a common coordinate system. Matching of the two curves is achieved when the discrepancy measurement is within a preset tolerance. This is done by adjusting the control parameters of the simulation model.

According to one embodiment, the discrepancy measurement is a function of distances between corresponding points of two curves. According to another embodiment, the discrepancy measurement includes a volume component under each segment of the pair of adjacent points.

DETAILED DESCRIPTION

The present invention pertains to methods and systems for matching a computed curve to a target curve to enable the determination of input variables to a computed-aided engineering analysis model. An example is the determination of the properties (constants) of a material model used within a computer-aided engineering analysis model. Discrepancies between computed curve and the target curve are measured, and based on the discrepancies, decisions on how to adjust parametric inputs can be made to achieve an optimal result of simulation. One exemplary target curve100is shown inFIG. 1. The curve100is a typical stress versus stretch-ratio (σ-λ) relationship obtained in a specimen test of polymeric material. Stretch-ratio (λ) is another form to define strain (ε). The relationship is defined as λ=1+ε. Curve100exhibits an hysteretic behavior. The σ-λ relationship starts at an original state102(i.e., zero stress and stretch-ratio of one (no stretch)) following a first load path112ato a second state104. Unloading from the second state104back to the original state102follows a second path112b. Reloading from the original state102to a third state106via the second state is through third and fourth paths112c-d. From the third state106, unloading back to the original state102is through a fifth path112e. Finally, another direct reloading from the original state102to the third state106is via a sixth path112f.

According to one aspect of the invention, one objective is to use a computer application to create a computed curve202that best matches the target curve204. To show how this is done,FIG. 2illustrates a computed curve202and a target curve204. The target curve204, Curve a, is bounded in a range from Xminto Xmaxand from Yminto Ymax. The computed curve202, Curve b, has a bounded range from Fminto Fmaxand from Gminto Gmax. It is evident that the two curves do not have the same range in abscissa. Matching them using prior art approaches could be very difficult.

Referring now toFIG. 3, it is shown a computed302and target curves304in a common normalized coordinate (ξ-η) system using a mapping scheme, according to an embodiment of the present invention. Curve A302, a normalized target curve, is defined by a first set of points (“m” points, shown as i=1, 2, . . . m, denoted by “|”). Curve B304, a normalized computed curve, is defined by a second set of points (“n” points, shown as j=1, 2, . . . n, denoted by “X”) with a third set of points (“m” points shown as p=1, 2, . . . m, denoted by circle) mapped thereon. A segment is defined by a straight line between two adjacent points. For example, δSi312is a segment of the target curve302, while δTp314is a segment of the computed curve304.

The mapping scheme is summarized in the following procedure:1) Normalize coordinates i of the first set of points of target curve a (FIG. 2) to its smallest
bounding ranges to create Curve A302.

ξi=Xi-XminXmax-Xmin;ηi=Yi-YminYmax-Ymin⁢⁢Xmin=mink⁢Xk;Xmax=maxk⁢Xk⁢⁢Ymin=mink⁢Yk;Ymax=maxk⁢Yk(1)2) Normalize coordinates j of the second set of points of computed curve b (FIG. 2) to its smallest bounding ranges to create Curve B304.

ξj′=Fi-FminFmax-Fmin;ηj′=Gi-GminGmax-Gmin⁢⁢Fmin=mink⁢Fk;Fmax=maxk⁢Fk⁢⁢Gmin=mink⁢Gk;Gmax=maxk⁢Gk(2)3) Compute S, the total length of Curve A304. Also compute the individual segment lengths δSi314:
δSi=√{square root over ((ξi−ξi-1)2+(ηi−ηi-1)2)}{square root over ((ξi−ξi-1)2+(ηi−ηi-1)2)};i=2,3, . . . ,m(3)
A segment is defined as a part of the curve between two adjacent points, connected by a straight line.4) Create a length ratio of each segment to the total length S:
{tilde over (s)}i=δSi/S;i=2,3, . . . ,m(4)5) Calculate total length T of Curve B304.6) Map each point on Curve A302to Curve B304using the following formula:
{tilde over (s)}i×T=δTp;i=p=2,3, . . . ,m(5)
A typical segment on Curve B304corresponds to a segment i of Curve A302has length δTp314.7) Denormalize the newly mapped computed curve by transforming it to its original coordinate system, but now represented by the third set of points which are mapped from the target curve.
xp=Fmin+ξi′(Fmax−Fmin);yp=Gmin+ηi′(Gmax−Gmin)  (6)8) Transform the resulting curve to the same coordinate system (i.e., a common coordinate system) as the normalized target curve A302. The third set of points now have coordinates as follows:

The two curves are now in a common coordinate system and can be presented in the same graph as shown inFIG. 3. With the initial point of both curves positioned at the origin, the discrepancy between the two curves can be obtained by comparing the point pairs (i.e., (ξ, η)310and (ξ″, η″)320).

According to one embodiment of the invention, the distance330between each point pair is calculated as:
di=√{square root over ((ξp″−ξi)2+(ηp″−ηi)2)}{square root over ((ξp″−ξi)2+(ηp″−ηi)2)}  (8)
and the volume component between the two curves is quantified by the sum of the areas between each segment pairs as

The discrepancy between two curves can be measured in a number of manners, for example, sum of the volume components, maximum of all the components, and alike.

Because the points used to calculate the discrepancies between the target and the computed curves are paired, there is no requirement for the abscissa to be monotonic. Open curves with hysteretic characteristics can also be matched without difficulties. The calculated discrepancy is linear with respect to the curve parameters when a large number of curve points are used. Therefore, the parameter identification of a linear system can be done in single step. Also, because the comparison of the two curves are based on a one-to-one mapping of the points on the target curve on a common coordinate system, even if the computed curve is completely set apart from the target curve, the discrepancy calculation from the transformed and normalized point pairs provides a reliable and robust measurement. The unstable computational in the traditional methods is avoid.

As the discrepancy between the computed and the target curves is determined, the optimization of the simulation can be performed by adjusting the parameters of the CAE model such that the discrepancy is minimized within a preset tolerance.

FIG. 4Ais a flowchart showing an exemplary optimization process400for matching a computed curve to a target curve using a computer aided engineering analysis model to enable realistic engineering simulation, according to an embodiment of the present invention. Process400is preferably implemented in software.

Process400starts by receiving a target curve at step402. The target curve is defined by a first set of points (e.g., “m” points shown inFIG. 3). One exemplary target curve is strain-stress relationship curve obtained in a material specimen test (e.g.,FIG. 1). At step404, a computer aided engineering (CAE) model (e.g., finite element analysis (FEA) model) is defined for matching the target curve. The CAE model includes a number of control adjustable parameters that can be adjusted as a means for matching a computed curve to the target curve. The computed curve is first obtained using the CAE model with the initial set of parameters at step408. The computed curve is defined by a second set of points (e.g., “n” points shown inFIG. 3). Generally, there is no requirement as to how the first set of points and the second set of points relate to each other. In other words, “m” and “n” can be different. Further, there is no requirement as to how the first and second sets of points are distributed, for example, points can be arbitrarily distributed on their respective curves.

Next, a discrepancy is measured between the computed and target curves using a mapping scheme at step410. Details of the mapping scheme are described inFIG. 4Band corresponding descriptions below. At decision412, it is determined whether the discrepancy is below a tolerance. If not, the parameters of the CAE model are adjusted at step414for creating another computed curve. Process400moves back to step408to repeat until decision412becomes true. Process400ends thereafter.

FIG. 4Bshows the details the mapping scheme of step410for calculating the discrepancy between the computed and the target curves. At410a, the smallest bounding ranges of the target and the computed curves are determined (FIG. 2). At410b, the first and second sets of points on the target and on the computed curves are normalized to their respective bounding ranges according to Equations (1) and (2) respectively. At410c, the length of each segment defined by two adjacent points on the target curve is calculated (Equation (3)) and accordingly the total length S of the target curve is obtained by summing all the segment length. The total length T of the computed curve is also calculated. At410d, a length ratio of each segment to the total length of the target curve is calculated (Equation (4)). At step410e, the third set of points is created using the length ratio according to Equation (5) on the computed curve. At410f, the third set of points on the computed curve are converted back to the original coordinate system (Equation (6)) and then normalized to the bounding ranges of the target curve as shown in Equation (7). Discrepancy is calculated using the point pairs between the first and third sets of points according to Equations (8)-(9) at410g. The calculated discrepancy is then carried to decision412ofFIG. 4A.

According to one embodiment, the present invention is directed towards one or more computer systems capable of carrying out the functionality described herein. An example of a computer system500is shown inFIG. 5. The computer system500includes one or more processors, such as processor504. The processor504is connected to a computer system internal communication bus502. Various software embodiments are described in terms of this exemplary computer system. After reading this description, it will become apparent to a person skilled in the relevant art(s) how to implement the invention using other computer systems and/or computer architectures.

Computer system500also includes a main memory508, preferably random access memory (RAM), and may also include a secondary memory510. The secondary memory510may include, for example, one or more hard disk drives512and/or one or more removable storage drives514, representing a floppy disk drive, a magnetic tape drive, an optical disk drive, etc. The removable storage drive514reads from and/or writes to a removable storage unit518in a well-known manner. Removable storage unit518, represents a floppy disk, magnetic tape, optical disk, etc. which is read by and written to by removable storage drive514. As will be appreciated, the removable storage unit518includes a computer usable storage medium having stored therein computer software and/or data.

In alternative embodiments, secondary memory510may include other similar means for allowing computer programs or other instructions to be loaded into computer system500. Such means may include, for example, a removable storage unit522and an interface520. Examples of such may include a program cartridge and cartridge interface (such as that found in video game devices), a removable memory chip (such as an Erasable Programmable Read-Only Memory (EPROM), Universal Serial Bus (USB) flash memory, or PROM) and associated socket, and other removable storage units522and interfaces520which allow software and data to be transferred from the removable storage unit522to computer system500. In general, Computer system500is controlled and coordinated by operating system (OS) software, which performs tasks such as process scheduling, memory management, networking and I/O services.

There may also be a communications interface524connecting to the bus502. Communications interface524allows software and data to be transferred between computer system500and external devices. Examples of communications interface524may include a modem, a network interface (such as an Ethernet card), a communication port, a Personal Computer Memory Card International Association (PCMCIA) slot and card, etc. Software and data transferred via communications interface524. The computer500communicates with other computing devices over a data network based on a special set of rules (i.e., a protocol). One of the common protocols is TCP/IP (Transmission Control Protocol/Internet Protocol) commonly used in the Internet. In general, the communication interface524manages the assembling of a data file into smaller packets that are transmitted over the data network or reassembles received packets into the original data file. In addition, the communication interface524handles the address part of each packet so that it gets to the right destination or intercepts packets destined for the computer500. In this document, the terms “computer program medium”, “computer readable medium”, “computer recordable medium” and “computer usable medium” are used to generally refer to media such as removable storage drive514(e.g., flash storage drive), and/or a hard disk installed in hard disk drive512. These computer program products are means for providing software to computer system200. The invention is directed to such computer program products.

Computer programs (also called computer control logic) are stored as application modules506in main memory508and/or secondary memory510. Computer programs may also be received via communications interface524. Such computer programs, when executed, enable the computer system500to perform the features of the present invention as discussed herein. In particular, the computer programs, when executed, enable the processor504to perform features of the present invention. Accordingly, such computer programs represent controllers of the computer system500.

In an embodiment where the invention is implemented using software, the software may be stored in a computer program product and loaded into computer system500using removable storage drive514, hard drive512, or communications interface524. The application module506, when executed by the processor504, causes the processor504to perform the functions of the invention as described herein.

The main memory508may be loaded with one or more application modules506that can be executed by one or more processors504with or without a user input through the I/O interface530to achieve desired tasks. In operation, when at least one processor504executes one of the application modules506, the results are computed and stored in the secondary memory510(i.e., hard disk drive512). The status of the analysis (e.g., computed curve) is reported to the user via the I/O interface530either in a text or in a graphical representation upon user's instructions.

Although the present invention has been described with reference to specific embodiments thereof, these embodiments are merely illustrative, and not restrictive of, the present invention. Various modifications or changes to the specifically disclosed exemplary embodiments will be suggested to persons skilled in the art. For example, whereas two-dimensional curves have been shown and described, other types of curves may be used instead, for example, three-dimensional curves. Additionally, the calibration of material properties has been shown and described, other types of system can be used instead, for example, a target curve representing a desired system behavior to be matched by tweaking certain design parameters. Furthermore whereas the steps for matching the two curves has been shown and described as a group of equations (Equations (1)-(9)), other equivalent mathematical descriptions of material behaviors can be used instead. In summary, the scope of the invention should not be restricted to the specific exemplary embodiments disclosed herein, and all modifications that are readily suggested to those of ordinary skill in the art should be included within the spirit and purview of this application and scope of the appended claims.