The invention generally relates to methods, systems and software product used in the area of computer-aided engineering analysis, more particularly to methods and systems used in obtaining numerically simulated structural behaviors of a product modeled with first order non-“volumetric-locking” (FONVL) finite elements.
With advance of computer technology, computer aided engineering (CAE) and computer aided design (CAD) have been used for helping engineers/scientists to design products in various industries (e.g., automotive, aerospace, etc.). One of the first developed CAE technologies is finite element analysis (FEA), which is a computerized method widely used in industry to model and solve engineering problems relating to complex systems such as three-dimensional non-linear structural design and analysis. FEA derives its name from the manner in which the geometry of the object under consideration is specified.
The FEA software is provided with a model of the geometric description and the associated material properties at each point within the model (sometimes referred to as a FEA mesh model). In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are referred to as finite elements. The vertices of the finite elements are referred to as nodes. The model is comprised of a finite number of finite elements, which are assigned a material name to associate with material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
In CAD, the outer surface of a physical object/product is represented by a three-dimensional wire frame of a number of connected triangles (i.e., polygons). To represent the entire volume of the physical object, tetrahedrons are generated from the surface model. Traditionally, the model automatically generated by CAD cannot be used for FEA due to several problems. For example, with the use of the piecewise linear shape function, the traditional triangular and tetrahedral elements are computationally efficient with one Gauss integration point. This linear shape function is also desirable for most contact algorithms. However, the well-known volumetric locking problem limits the use of these simple elements for general FEA stress analysis.
To solve the volumetric locking problem, one prior art approach uses higher order elements. However, this may be used in static stress analysis. Non-linear shape function would cause a time-step issue in dynamic analysis using explicit solver. Contact algorithm would also become very costly due to curved contact surfaces.
Another prior art approach uses cubic bubble functions as shape functions. However, this would cause other shortcomings such as losing Kronecker delta property in shape functions, thereby defeating the purpose of using FEA.
It would therefore be desirable to have improved methods and systems for numerically simulating structural behaviors of a product modeled with finite elements without volumetric locking.
This section is for the purpose of summarizing some aspects of the present application and to briefly introduce 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 application.
Systems and methods for numerically simulating structural behaviors of a product modeled with first order non-“volumetric-locking” (FONVL) finite elements are disclosed. According to one aspect, a computerized model (e.g., FEA model) representing a product is received in a computer system having a FEA application module installed thereon. FEA model contains nodal points connected by FONVL finite elements. Each FONVL finite element is configured with volumetric DOFs at respective boundaries. The volumetric DOFs are configured such that (a) shape functions of each FONVL finite element stay the same, and (b) shear stresses of each FONVL finite element stay constant.
Numerically-simulated structural behaviors of the product are then obtained by conducting a time-marching simulation using the FEA model. Time-marching simulation covers a time duration in a number of solution cycles. At each solution cycle for each FONVL finite element, calculating volumetric variations at the respective boundaries based on corresponding one of the volumetric DOFs; and then solving unknown coefficients of a linear formula that defines said each FONVL finite element's volumetric strain distribution using the calculated volumetric variations along with representative coordinates of the respective boundaries.
Objects, features, and advantages of the invention will become apparent upon examining the following detailed description of an embodiment thereof, taken in conjunction with the attached drawings.
These and other features, aspects, and advantages of the invention will be better understood with regard to the following description, appended claims, and accompanying drawings as follows:
In the following description, numerous specific details are set forth in order to provide a thorough understanding of embodiments of the invention. However, it will become obvious to those skilled in the art that the invention may be practiced without these specific details. The descriptions and representations herein are the common means used by those experienced or skilled in the art to most effectively convey the substance of their work to others skilled in the art. In other instances, well-known methods, procedures, and components have not been described in detail to avoid unnecessarily obscuring aspects of the invention.
Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments. Further, the order of blocks in process flowcharts or diagrams representing one or more embodiments of the invention do not inherently indicate any particular order nor imply any limitations in the invention.
Embodiments of the invention are discussed herein with reference to
Referring first to
Process 200 starts at step 202 by receiving a computerized model (e.g., finite element analysis (FEA) mesh model) representing a product in a computer system (e.g., computer system 600 shown in
The following equations are for the two-dimensional and three-dimensional FONVL finite elements:
For a 2D FONVL finite element, the linear shape functions are defined as:
N
(1)=ξ1
N
(2)=ξ2 (1)
N
(3)=1−ξ1−ξ2
For a 3D FONVL finite element, the linear shape functions are defined as:
N
(1)=ξ1
N
(2)=ξ2 (2)
N
(3)=ξ3
N
(4)=1−ξ1ξ2−ξ3
where (ξ1, ξ2) and (ξ1, ξ2, ξ3) are the 2D and 3D FONVL finite element parameters, respectively. An example shape function 351 for a 2D FONVL finite element is shown in
The Jacobian for the 2-D FONVL finite element is as follows:
The Jacobian for the 3-D FONVL finite element is computed as follows:
where xi(n) are the nodal coordinates in a global coordinate system (e.g., global coordinate system 310 of
The strains of 2-D FONVL finite element are computed as follows:
The strains of 3-D FONVL finite element are computed as follows:
where ui(n) are the nodal displacements in a global coordinate system.
Using Equations (5) and (6), average volumetric strain for 2D FONVL finite element is defined as:
Average volumetric strain for 3D FONVL finite element is defined as:
Pure shear strains for 2D FONVL finite element are computed as:
where i and j=1, 2 are indices for dimensions (i.e., 2 for 2D).
Pure shear strains for 3D FONVL finite element are computed as:
where i and j=1, 3 are indices for dimensions (i.e., 3 for 3D) and δif is the Kronecker delta tensor.
Referring back to process 200, at step 204, numerically-simulated structural behaviors of the product in response to a loading are obtained by conducting a time-marching simulation (i.e., time-domain non-linear structural dynamic analysis based on FEA) using the FEA mesh model. The time-marching simulation covers a time duration in a number of solution cycles or time steps. At each solution cycle for each FONVL finite element, volumetric variations at the respective boundaries are calculated based on the value of corresponding volumetric DOF. Unknown coefficients of a linear formula that defines said each FONVL finite element's volumetric strain distribution are then solved using the calculated volumetric variations along with representative coordinates of the respective boundaries (e.g., at centroids of the boundaries).
The linear formula for volumetric strain distribution is as follows:
where α0 and αi are unknown coefficients. For a 2D FONVL finite element, there are three unknown coefficients and three volumetric DOFs. For 3D FONVL finite element, there are four unknown coefficients and four volumetric DOFs.
In order to solve unknown coefficients in the linear formula, the same number of independent equations needs to be established. One method is to determine the volumetric strain at each of the volumetric DOFs.
For 2D FONVL finite element, the volumetric strain at each of the volumetric DOFs is
where V0 is the original area of the 2D FONVL finite element, ΔV(m) is the volumetric variation at one of the boundaries with respect to corresponding volumetric DOF.
For 3D FONVL finite element, the volumetric strain at each of the volumetric DOFs is
where V0 is the original volume of the 3D FONVL finite element, ΔV(m) is the volumetric variation at one of the boundaries with respect to corresponding volumetric DOF.
The volumetric variation at one of the boundaries with respect to corresponding volumetric DOF in a 2D FONVL finite element is calculated as follows:
ΔV(m)=½L(m)h(m), m=1,3 (15)
where L(m) is the length of the boundary and h(m) is the value of the corresponding volumetric DOF. In other words, the first example volumetric variation shown in
In another embodiment, the second example volumetric variation shown in
The volumetric variation at one of the boundaries corresponding to the volumetric DOF in a 3D FONVL finite element is calculated as follows:
ΔV(m)=⅓A(m)h(m), m=1,4 (16)
where A(m) is the area of the face or boundary and h(m) is the value of the corresponding volumetric DOF.
Other schemes for calculating volumetric variations may be used in 3D FONVL finite element similar to the one shown in
Since the linear formula contains the same number of unknown coefficients as the number of volumetric DOFs in each FONVL finite element, hence the solution can be found by forming a set of simultaneous independent equations at centroids of the boundaries (i.e., edges for 2D and faces for 3D FONVL finite element). The set of simultaneous equation uses corresponding coordinates of the centroids along with the volumetric strain values calculated in Equation (13) or Equation (14) for 2D FONVL finite element or 3D FONVL finite element.
An embodiment of the invention is directed towards one or more computer systems capable of carrying out the functionality described herein. An example of a computer system 600 is shown in
Computer system 600 also includes a main memory 608, preferably random access memory (RAM), and may also include a secondary memory 610. The secondary memory 610 may include, for example, one or more hard disk drives 612 and/or one or more removable storage drives 614, representing a floppy disk drive, a magnetic tape drive, an optical disk drive, etc. The removable storage drive 614 reads from and/or writes to a removable storage unit 618 in a well-known manner. Removable storage unit 618, represents a floppy disk, magnetic tape, optical disk, etc. which is read by and written to by removable storage drive 614. As will be appreciated, the removable storage unit 618 includes a computer usable storage medium having stored therein computer software and/or data.
In alternative embodiments, secondary memory 610 may include other similar means for allowing computer programs or other instructions to be loaded into computer system 600. Such means may include, for example, a removable storage unit 622 and an interface 620. 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 units 622 and interfaces 620 which allow software and data to be transferred from the removable storage unit 622 to computer system 600. In general, Computer system 600 is 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 interface 624 connecting to the bus 602. Communications interface 624 allows software and data to be transferred between computer system 600 and external devices. Examples of communications interface 624 may include a modem, a network interface (such as an Ethernet card), a communications port, a Personal Computer Memory Card International Association (PCMCIA) slot and card, etc. Software and data transferred via communications interface 624 are in the form of signals which may be electronic, electromagnetic, optical, or other signals capable of being received by communications interface 624. The computer 600 communicates 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 interface 624 manages 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 interface 624 handles the address part of each packet so that it gets to the right destination or intercepts packets destined for the computer 600. In this document, the terms “computer program medium”, “computer usable medium”, and “computer readable medium” are used to generally refer to media such as removable storage drive 614, and/or a hard disk installed in hard disk drive 612. These computer program products are means for providing software to computer system 600. The invention is directed to such computer program products.
The computer system 600 may also include an input/output (I/O) interface 630, which provides the computer system 600 to access monitor, keyboard, mouse, printer, scanner, plotter, and alike.
Computer programs (also called computer control logic) are stored as application modules 606 in main memory 608 and/or secondary memory 610. Computer programs may also be received via communications interface 624. Such computer programs, when executed, enable the computer system 600 to perform the features of the invention as discussed herein. In particular, the computer programs, when executed, enable the processor 604 to perform features of the invention. Accordingly, such computer programs represent controllers of the computer system 600.
In an embodiment where the invention is implemented using software, the software may be stored in a computer program product and loaded into computer system 600 using removable storage drive 614, hard drive 612, or communications interface 624. The application module 606, when executed by the processor 604, causes the processor 604 to perform the functions of the invention as described herein.
The main memory 608 may be loaded with one or more application modules 606 that can be executed by one or more processors 604 with or without a user input through the I/O interface 630 to achieve desired tasks. In operation, when at least one processor 604 executes one of the application modules 606, the results are computed and stored in the secondary memory 610 (i.e., hard disk drive 612). The status of the computer simulation (e.g., finite element analysis results) is reported to the user via the I/O interface 630 in either text or graphical representation.
Although the invention has been described with reference to specific embodiments thereof, these embodiments are merely illustrative, and not restrictive of, the invention. Various modifications or changes to the specifically disclosed exemplary embodiments will be suggested to persons skilled in the art. For example, whereas most of the figures in this document are for 2D FONVL finite element for illustration clarity and simplicity, the invention is also for 3D FONVL finite element. Furthermore, term “volumetric variation” has been used and shown throughout this document. “Volumetric variation” is actually area difference in a 2D FONVL finite element while “volumetric variation” is volume difference in a 3D FONVL finite element. Finally, term “boundary” or “boundaries” have been used and shown throughout. “Boundary” or “boundaries” are referred to as edge(s) in a 2D FONVL finite element and referred to as face(s) in a 3D FONVL finite element. 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.