Engineering Notes ENGINEERING NOTES are short manuscripts describing new developments or importan... more Engineering Notes ENGINEERING NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes cannot exce ed 6 manuscript pages and 3 figures; a page of text may be substiru redfor a figure and vice versa. After informal review by the editors, they may be published wttã few months of the dare of receipt. Style requirements are the same as for regular contributions (see inside back cover).
In the present study, inertial forces and kinematics
of wings of locusts, Schistocerca americana,... more In the present study, inertial forces and kinematics of wings of locusts, Schistocerca americana, were investigated experimentally. The developed experimental setup includes freshly extracted hindwings and forewings, a mechanical transmission producing pitching and flapping, a vacuum chamber, and a high-speed video system. Flapping angle amplitudes determined at the middle section are practically the same in air and in vacuum in both forewings and hindwings. Pitching amplitudes in the root and in the middle section of hindwing differ by up to 50 % due to torsional deformation or twist. In air, the average twist angles are 3° and 17° in forewings and hindwings, respectively. Amplitudes of twist angle are higher in vacuum, especially at relatively high flapping frequencies. Inertial forces are calculated based on a rigidbody model of a wing with prescribed displacements. The 5- segment mesh for hindwings and 2-segment mesh for forewings provide accurate determination of inertial forces. Amplitudes of the fifth harmonic in accelerations in air are substantially smaller than in vacuum and, therefore, inertial forces in air are about 30 % smaller than those in vacuum. Thus, aerodynamic loads suppressed high-frequency component in wing oscillations resulting in substantially smaller inertial forces in air.
Aerodynamic and inertial forces and corresponding kinematics of flapping wings of locusts,
Schist... more Aerodynamic and inertial forces and corresponding kinematics of flapping wings of locusts, Schistocerca americana, were investigated in a low-speed wind tunnel. The experimental setup included live locusts mounted on microbalance synchronized with a high-speed video system. Simultaneous measurements of wing kinematics and forces were carried out on three locusts at 7° angle of attack and velocities of 0ms−1 and 4ms−1. Time variations of flapping and pitching angles exhibit similar patterns in fore- and hindwings and among the animals. Significant tip to root variations in pitching angle are found in both wings. The locusts have much larger flapping and pitching amplitudes in still air causing larger oscillations in inertial forces. Inertial forces are added to the lift and thrust on one part of the stroke, resulting in higher reaction forces and subtracted on the other part. Plots of the lift demonstrate similar trends with and without the wind. The global maxima and peak-to-peak amplitudes in lift are about the same in both tests. However, local minima are significantly lower in still air, resulting in much smaller stroke-averaged lift. Amplitudes of thrust force oscillations are much higher in still air; consequently, the stroke-averaged thrust is higher compared to the non-zero freestream velocity case.
Effects of the sideslip angle on the wing kinematics and the aerodynamic forces and
moments gener... more Effects of the sideslip angle on the wing kinematics and the aerodynamic forces and moments generated by the locusts were investigated in a low-speed wind tunnel. Three live locusts (Schistocerca americana) were tested at tunnel speeds of 0, 2, and 4 m/s at three body angles of attack (0°, 3° and 7°) and three sideslip angles (0°, -10° and -20°). Asensitive custom-built microbalance was used to measure the forces and moments. The balance was synchronized with high-speed video system recording the wing kinematics. It was observed that the nose down pitching moment increases with the angle of attack increase. This is an indication of active and/or passive mechanisms restoring the body orientation in the vertical plane and providing pitch stability of locusts. The side airflow generates a large side force and positive rolling moment. In response to the oncoming side wind, the locust executes larger-amplitude flapping motion on the windward wings. Changes in kinematic parameters of locust wings are similar to those observed in dragonfly during the right-turn maneuver. However, the yawing moment increases with the sideslip angle increase indicating yaw stability. This could be due to the weathercock stability, since about two thirds of locust’s body length and legs lie behind its center of mass.
This study provides a literature review relevant to bio-inspired micro air vehicle design, with e... more This study provides a literature review relevant to bio-inspired micro air vehicle design, with emphasis placed on flapping locust wings. General ornithopter design and fabrication options are considered. The construction of four artificial wings is described, they have 49 mm spans and varied chordwise stiffnesses. Two sets of locust hindwings are extracted, tested and compared with artificial wings. Tests are performed at zero freestream velocity on two different transmissions, one strictly flapping and the other a combination of active flapping and pitching. Frequency is varied between 10 and 20 Hz. Simultaneous video and force data is acquired; kinematic and aerodynamic results are discussed, both stroke-averaged and instantaneous.
Aerodynamic forces and moments produced by locusts (Schistocerca americana) were investigated in ... more Aerodynamic forces and moments produced by locusts (Schistocerca americana) were investigated in a low-speed wind tunnel. The experimental setup included locusts mounted on a sensitive six-component microbalance using a sting. Tests were conducted at tunnel speeds of 0, 2, 3, 3.5 and 4 m/s at two body angles of attack and five positive yaw angles at each body angle of attack. The experimental results are presented for stroke-averaged forces and moments normalized using the insect body weight and length. Locusts generated more lift at higher velocities and at higher angles of attack, but none of locust generated lift greater than its weight. They generally produced net thrust at lower velocities and lower angles of attack and net drag at higher velocities, as expected. Interestingly, locusts responded to yaw by generating more positive side force, yawing moment and negative rolling moment. The lift force and pitching moment were largely insensitive to yaw.
Aerodynamic characteristics of freshly isolated hind-wings of locusts and synthetic wing models w... more Aerodynamic characteristics of freshly isolated hind-wings of locusts and synthetic wing models were investigated in the present study. The experimental setup included a pair of wings mounted atop a flapping-pitching device executing wings motion through a transmission shaft. The resulting kinematics consists of flapping with amplitude of 85º and pitching about the wing axis with amplitude of 60º. The device was mounted on a custom 6-component microbalance to measure the forces and moments generated by the flapping wings. Model wings made of balsa wood, composite materials and locust wings were tested in still air at flapping frequencies ranging from 5 Hz to 30 Hz. The experimental results are presented for stroke-averaged thrust and lift as well as respective coefficients averaged over several consecutive flapping cycles. Aerodynamic efficiency of insect wings and man made wings are compared and commented on.
Effects of the sideslip angle on the wing-kinematics and the aerodynamic forces and moments gener... more Effects of the sideslip angle on the wing-kinematics and the aerodynamic forces and moments generated by the locusts were investigated in a low-speed wind tunnel. Three locusts (Schistocerca americana) were tested at tunnel speeds of 0, 2, and 4 m/s at three body angles of attack (0, 3 and 7 degrees) and three sideslip angles (0, -10 and -20 degrees) at each angle of attack. A sensitive custom-built microbalance was used to measure the forces and moments. Simultaneous high - speed videos were taken to record the wing kinematics. Results in the form of normalized stroke-averaged forces and moments are presented and a correlation with the wing kinematics has been investigated.
Instantaneous forces on flapping wings are influenced by inertia of moving masses of the wings. I... more Instantaneous forces on flapping wings are influenced by inertia of moving masses of the wings. In the present study, inertial forces and kinematics of wings of locusts, Schistocerca americana, were investigated experimentally. The developed experimental setup includes freshly extracted hindwings and forewings, a mechanical transmission producing pitching and flapping, a vacuum chamber, and a high-speed video system. Masses and locations of mass centers were measured and averaged based on data for four locusts. Videos were taken and time-resolved displacements of markers on flapping-pitching wings were obtained. Flapping angle amplitudes determined at the middle section are practically the same in air and in vacuum in both forewings and hindwings. Pitching amplitudes in the root and in the middle section of hindwing differ by up to 50% due to torsional deformation or twist. In air, the average twist angles are 3 and 18 in forewings and hindwings, respectively. Amplitudes of twist angle are higher in vacuum, especially at relatively high flapping frequencies. Inertial forces are calculat ed based on a rigid-body model of a wing with prescribed displacements. The 5-segment mesh for hindwings and 2-segment mesh for forewings provide accurate determination of inertial forces. Vertical components of inertial forces are much higher than horizontal ones because the flapping motion takes place in the vertical plane. The same forewings and hindwings were tested in air and in vacuum. Inertial forces in air are substantially smaller than in vacuum, because amplitudes of the fourth and fifth harmonics in wing accelerations in air are substantially smaller. Thus, aerodynamic forces reduced high-frequency oscillations resulting in smaller inertial forces in air. Overall, the aerodynamic damping is an important factor causing changes in wing kinema tics and consequently in inertial forces generated in flapping wings.
Aerodynamic and inertial forces and corresponding kinematics of flapping wings of locusts, Schist... more Aerodynamic and inertial forces and corresponding kinematics of flapping wings of locusts, Schistocerca Americana, were investigated in a low-speed wind tunnel. The experimental setup included live locusts mounted on microbalance synchronized with a high-speed video system. Simultaneous measurements of wing kinematics and forces were carried out on the three locusts at zero velocity and at 4 m/s and 7° angle of angle of attack. Time variations of flapping and pitching angles exhibit similar patterns in fore- and hindwings and among animals. Significant tip to root variations in pitching angle are found in both wings. The locusts have much larger flapping and pitching amplitudes in still air causing larger oscillations in inertial forces. Inertial forces are added to the lift and thrust on one part of the stroke, resulting in higher reaction forces and subtracted on the other part. Plots of the lift demonstrate similar trends with and without the wind. The global maxima and peak-to-peak amplitudes in lift are about the same in both tests. However, local minima are significantly lower at zero freestream, resulting in much smaller stroke-averaged lift. Amplitudes of thrust force oscillations are much higher at zero freestream, consequently, the stroke-averaged thrust is much higher compared to the non-zero freestream velocity case.
This study provides a literature review relevant to bio-inspired micro air vehicle design, with e... more This study provides a literature review relevant to bio-inspired micro air vehicle design, with emphasis placed on flapping locust hindwings. General ornithopter design and fabrication options are considered. Three sets of artificial wings are designed, constructed and tested. The artificial wings have varied chordwise stiffness and are each 49 mm in length. Locust wing kinematics are quantified and used to design two different transmissions: one strictly flapping and the other a combination of active pitching and flapping. Force data are acquired for the locust and artificial wings; stroke-averaged results are discussed. Flapping-wing experiments span 10 to 26 Hz at zero freestream velocity. Lastly, a 14-cm wingspan radio-controlled ornithopter is designed and built using some considered concepts. The aircraft was successfully tested and it performed controlled flight and demonstrated an ability to withstand moderate winds.
The least squares finite element method (LSFEM) which
is based minimizing the l2-norm of the resi... more The least squares finite element method (LSFEM) which is based minimizing the l2-norm of the residuals is regarded as an alternative approach to the well known Galerkin finite element methods (GFEM). In a variational setting, the LSFEM, unlike GFEM leads to minimization problem where compatibility conditions between approximation spaces like the restrictive LBB condition never arises and the re sulting linear algebraic system has a symmetric positive definite matrix. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have limited the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows. A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection -dominated flows and allows for equal-order basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with results of several unsteady incompressible viscous flow benchmark problems solved using low-order C0 continuous elements.
The least-squares finite element method (LSFEM), which
is based on minimizing the l2-norm of the ... more The least-squares finite element method (LSFEM), which is based on minimizing the l2-norm of the residual, has many attractive advantages over Galerkin finite element method (GFEM). It is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike GFEM. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have limited the application of LSFEM to large-scale practical problems, such as three -dimensional compressible viscous flows. A simple finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by pure a LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal- order basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with preliminary results of several benchmark problems solved using low-order C0 continuous elements.
A novel finite element method is proposed that employs a
least-squares method for first-order der... more A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated in the context of compressible flows by results of few compressible benchmark problems solved using low- order C0 continuous elements.
The least-squares finite element method (LSFEM), based on
minimizing the l2-norm of the residual ... more The least-squares finite element method (LSFEM), based on minimizing the l2-norm of the residual is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike Galerkin finite element method (GFEM). However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computational requirements that have limited the application of LSFEM to large-scale practical problems. A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The method has been successfully applied here to solve complex buoyancy-driven flow with Boussinesq approximation in a square cavity with differentially heated vertical walls using low-order C0 continuous elements.
The least-squares finite element method LSFEM has received increasing attention in recent years... more The least-squares finite element method LSFEM has received increasing attention in recent years due to advantages over the Galerkin finite element method GFEM. The method leads to a minimization problem in the L2-norm and thus results in a symmetric and positive definite matrix, even for first-order differential equations. In addition, the method contains an implicit streamline upwinding mechanism that prevents the appearance of oscillations that are characteristic of the Galerkin method. Thus, the least-squares approach does not require explicit stabilization and the associated stabilization parameters required by the Galerkin method. A new approach, the bubble enriched least-squares finite element method BELSFEM, is presented and compared with the classical LSFEM. The BELSFEM requires a space-time element formulation and employs bubble functions in space and time to increase the accuracy of the finite element solution without degrading computational performance. We apply the BELSFEM and classical least-squares finite element methods to benchmark problems for 1D and 2D linear transport. The accuracy and performance are compared.
The collision avoidance behavior of the locust, Schistocerca americana, in response to simulated ... more The collision avoidance behavior of the locust, Schistocerca americana, in response to simulated approaching objects, also called looming stimuli, was investigated in a low speed wind tunnel. The animals were mounted using a sting on a sensitive six-component microbalance custom-designed for the experiments. Forces and moments were measured as a function of time during the simulated approach and interpreted in the context of collision avoidance behaviors. The stimuli presented from the side effectively evoked robust and discernible collision avoidance responses. Locusts attempted to avoid collision by either flying over or under the looming object, or alternatively by steering around it. These efforts appeared in the form of changes in aerodynamic forces and moments as a function of time. It was also observed that the locusts increased the wing flapping frequency in response to the stimulus. is of great interest for engineering as it points to robust visuo-motor control mechanisms . Furthermore, locusts possess in their brains collision sensitive neurons that are involved in the generation of collision avoidance responses. All these factors have contributed to the extensive use of insects, and especially locusts, in collision avoidance research. Wing movements are central to these studies as they generate the aerodynamic forces and moments required in these avoidance maneuvers. Evidence of strong linkages between these wing movements and neural activity has inspired research across disciplines from neurobiology to robotics and aerodynamics.
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Papers by Rajeev KUMAR
of wings of locusts, Schistocerca americana, were investigated
experimentally. The developed experimental setup includes
freshly extracted hindwings and forewings, a mechanical
transmission producing pitching and flapping, a vacuum
chamber, and a high-speed video system. Flapping angle amplitudes
determined at the middle section are practically the
same in air and in vacuum in both forewings and hindwings.
Pitching amplitudes in the root and in the middle section of
hindwing differ by up to 50 % due to torsional deformation or
twist. In air, the average twist angles are 3° and 17° in forewings
and hindwings, respectively. Amplitudes of twist angle
are higher in vacuum, especially at relatively high flapping
frequencies. Inertial forces are calculated based on a rigidbody
model of a wing with prescribed displacements. The 5-
segment mesh for hindwings and 2-segment mesh for forewings
provide accurate determination of inertial forces.
Amplitudes of the fifth harmonic in accelerations in air are
substantially smaller than in vacuum and, therefore, inertial
forces in air are about 30 % smaller than those in vacuum.
Thus, aerodynamic loads suppressed high-frequency component
in wing oscillations resulting in substantially smaller inertial
forces in air.
Schistocerca americana, were investigated in a low-speed wind tunnel. The experimental setup
included live locusts mounted on microbalance synchronized with a high-speed video system.
Simultaneous measurements of wing kinematics and forces were carried out on three locusts at 7°
angle of attack and velocities of 0ms−1 and 4ms−1. Time variations of flapping and pitching angles
exhibit similar patterns in fore- and hindwings and among the animals. Significant tip to root
variations in pitching angle are found in both wings. The locusts have much larger flapping and
pitching amplitudes in still air causing larger oscillations in inertial forces. Inertial forces are added to
the lift and thrust on one part of the stroke, resulting in higher reaction forces and subtracted on the
other part. Plots of the lift demonstrate similar trends with and without the wind. The global maxima
and peak-to-peak amplitudes in lift are about the same in both tests. However, local minima are
significantly lower in still air, resulting in much smaller stroke-averaged lift. Amplitudes of thrust
force oscillations are much higher in still air; consequently, the stroke-averaged thrust is higher
compared to the non-zero freestream velocity case.
moments generated by the locusts were investigated in a low-speed wind tunnel. Three
live locusts (Schistocerca americana) were tested at tunnel speeds of 0, 2, and 4 m/s at
three body angles of attack (0°, 3° and 7°) and three sideslip angles (0°, -10° and -20°).
Asensitive custom-built microbalance was used to measure the forces and moments. The
balance was synchronized with high-speed video system recording the wing kinematics.
It was observed that the nose down pitching moment increases with the angle of attack
increase. This is an indication of active and/or passive mechanisms restoring the body
orientation in the vertical plane and providing pitch stability of locusts. The side airflow
generates a large side force and positive rolling moment. In response to the oncoming
side wind, the locust executes larger-amplitude flapping motion on the windward wings.
Changes in kinematic parameters of locust wings are similar to those observed in
dragonfly during the right-turn maneuver. However, the yawing moment increases with
the sideslip angle increase indicating yaw stability. This could be due to the weathercock
stability, since about two thirds of locust’s body length and legs lie behind its center of
mass.
were conducted at tunnel speeds of 0, 2, 3, 3.5 and 4 m/s at two body angles of attack and five positive yaw angles at each body angle of attack. The experimental results are presented for stroke-averaged forces and moments normalized using the insect body weight and length. Locusts generated more lift
at higher velocities and at higher angles of attack, but none of locust generated lift greater than its weight. They generally produced net thrust at lower velocities and lower angles of attack and net drag at higher velocities, as expected. Interestingly, locusts responded to yaw by generating more positive side force, yawing moment and negative rolling moment. The lift force and pitching moment were largely insensitive to yaw.
2, and 4 m/s at three body angles of attack (0, 3 and 7 degrees) and three sideslip angles (0, -10 and -20 degrees) at each angle of attack. A sensitive custom-built microbalance was used to measure the forces and moments. Simultaneous high -
speed videos were taken to record the wing kinematics. Results in the form of normalized stroke-averaged forces and moments are presented and a correlation with the wing kinematics has been investigated.
americana, were investigated experimentally. The developed experimental setup includes freshly extracted hindwings and forewings, a mechanical transmission producing pitching
and flapping, a vacuum chamber, and a high-speed video system. Masses and locations of mass centers were measured and averaged based on data for four locusts. Videos were taken
and time-resolved displacements of markers on flapping-pitching wings were obtained. Flapping angle amplitudes determined at the middle section are practically the same in air and in vacuum in both forewings and hindwings. Pitching amplitudes in the root and in the middle section of hindwing differ by up to 50% due to torsional deformation or twist. In air, the average twist angles are 3 and 18 in forewings and hindwings, respectively. Amplitudes of twist angle are higher in vacuum, especially at relatively high flapping frequencies. Inertial forces are calculat ed based on a rigid-body model of a wing with prescribed
displacements. The 5-segment mesh for hindwings and 2-segment mesh for forewings provide accurate determination of inertial forces. Vertical components of inertial forces are
much higher than horizontal ones because the flapping motion takes place in the vertical plane. The same forewings and hindwings were tested in air and in vacuum. Inertial forces in
air are substantially smaller than in vacuum, because amplitudes of the fourth and fifth harmonics in wing accelerations in air are substantially smaller. Thus, aerodynamic forces reduced high-frequency oscillations resulting in smaller inertial forces in air. Overall, the aerodynamic damping is an important factor causing changes in wing kinema tics and consequently in inertial forces generated in flapping wings.
wing kinematics and forces were carried out on the three locusts at zero velocity and at 4 m/s and 7° angle of angle of attack. Time variations of flapping and pitching angles exhibit similar patterns in fore- and hindwings and among animals. Significant tip to root variations in pitching angle are found in both wings. The locusts have much larger flapping and pitching amplitudes in still air causing larger oscillations in inertial forces. Inertial forces are added to the lift and thrust on one part of the stroke, resulting in higher reaction forces and subtracted on the other part. Plots of the lift demonstrate similar trends with and without the wind. The global maxima and peak-to-peak amplitudes in lift
are about the same in both tests. However, local minima are significantly lower at zero freestream, resulting in much smaller stroke-averaged lift. Amplitudes of thrust force oscillations are much higher at zero freestream, consequently, the stroke-averaged thrust is much higher compared to the non-zero freestream velocity case.
designed and built using some considered concepts. The aircraft was successfully tested and it performed controlled flight and demonstrated an ability to withstand moderate winds.
is based minimizing the l2-norm of the residuals is regarded as
an alternative approach to the well known Galerkin finite
element methods (GFEM). In a variational setting, the
LSFEM, unlike GFEM leads to minimization problem where
compatibility conditions between approximation spaces like
the restrictive LBB condition never arises and the re sulting
linear algebraic system has a symmetric positive definite
matrix. However, the higher continuity requirements for
second-order terms in the governing equations force the
introduction of additional unknowns through the use of an
equivalent first-order system of equations or the use of C1
continuous basis functions. These additional unknowns lead to
increased memory and computing time requirements that have
limited the application of LSFEM to large-scale practical
problems, such as three-dimensional compressible viscous
flows. A novel finite element method is proposed that employs
a least-squares method for first-order derivatives and a
Galerkin method for second order derivatives, thereby
avoiding the need for additional unknowns required by a pure
LSFEM approach. When the unsteady form of the governing
equations is used, a streamline upwinding term is introduced
naturally by the least-squares method. Resulting system matrix
is always symmetric and positive definite and can be solved
by iterative solvers like pre-conditioned conjugate gradient
method. The method is stable for convection -dominated flows
and allows for equal-order basis functions for both pressure
and velocity. The stability and accuracy of the method are
demonstrated with results of several unsteady incompressible
viscous flow benchmark problems solved using low-order C0
continuous elements.
is based on minimizing the l2-norm of the residual, has many
attractive advantages over Galerkin finite element method
(GFEM). It is now well established as a proper approach to
deal with the convection dominated fluid dynamic equations.
The least-squares finite element method has a number of
attractive characteristics such as the lack of an inf-sup
condition and the resulting symmetric positive system of
algebraic equations unlike GFEM. However, the higher
continuity requirements for second-order terms in the
governing equations force the introduction of additional
unknowns through the use of an equivalent first-order system
of equations or the use of C1 continuous basis functions. These
additional unknowns lead to increased memory and computing
time requirements that have limited the application of LSFEM
to large-scale practical problems, such as three -dimensional
compressible viscous flows. A simple finite element method is
proposed that employs a least-squares method for first-order
derivatives and a Galerkin method for second order
derivatives, thereby avoiding the need for additional
unknowns required by pure a LSFEM approach. When the
unsteady form of the governing equations is used, a streamline
upwinding term is introduced naturally by the least-squares
method. Resulting system matrix is always symmetric and
positive definite and can be solved by iterative solvers like
pre-conditioned conjugate gradient method. The method is
stable for convection-dominated flows and allows for equal-
order basis functions for both pressure and velocity. The
stability and accuracy of the method are demonstrated with
preliminary results of several benchmark problems solved
using low-order C0 continuous elements.
least-squares method for first-order derivatives and a Galerkin
method for second order derivatives, thereby avoiding the
need for additional unknowns required by a pure LSFEM
approach. When the unsteady form of the governing equations
is used, a streamline upwinding term is introduced naturally
by the least-squares method. Resulting system matrix is
always symmetric and positive definite and can be solved by
iterative solvers like pre-conditioned conjugate gradient
method. The method is stable for convection-dominated flows
and allows for equal-order basis functions for both pressure
and velocity. The stability and accuracy of the method are
demonstrated in the context of compressible flows by results
of few compressible benchmark problems solved using low-
order C0 continuous elements.
minimizing the l2-norm of the residual is now well established
as a proper approach to deal with the convection dominated
fluid dynamic equations. The least-squares finite element
method has a number of attractive characteristics such as the
lack of an inf-sup condition and the resulting symmetric
positive system of algebraic equations unlike Galerkin finite
element method (GFEM). However, the higher continuity
requirements for second-order terms in the governing
equations force the introduction of additional unknowns
through the use of an equivalent first-order system of
equations or the use of C1 continuous basis functions. These
additional unknowns lead to increased memory and
computational requirements that have limited the application
of LSFEM to large-scale practical problems.
A novel finite element method is proposed that employs a
least-squares method for first-order derivatives and a Galerkin
method for second order derivatives, thereby avoiding the
need for additional unknowns required by a pure LSFEM
approach. When the unsteady form of the governing equations
is used, a streamline upwinding term is introduced naturally
by the least-squares method. Resulting system matrix is
always symmetric and positive definite and can be solved by
iterative solvers like pre-conditioned conjugate gradient
method. The method is stable for convection-dominated flows
and allows for equal-order basis functions for both pressure
and velocity. The method has been successfully applied here
to solve complex buoyancy-driven flow with Boussinesq
approximation in a square cavity with differentially heated
vertical walls using low-order C0 continuous elements.
minimization problem in the L2-norm and thus results in a symmetric and positive definite matrix, even for first-order differential equations. In addition, the method contains an implicit streamline upwinding mechanism that prevents the appearance of oscillations that are characteristic of the Galerkin method. Thus, the least-squares approach does not require explicit stabilization and the associated stabilization parameters required by the Galerkin method. A new approach, the bubble enriched least-squares finite element method BELSFEM, is presented and compared with the classical LSFEM. The BELSFEM requires a space-time element formulation and employs bubble functions in space and time to increase the accuracy of the finite element solution without degrading computational performance. We apply the BELSFEM and classical least-squares finite element methods to benchmark problems for 1D and 2D linear transport. The accuracy and
performance are compared.