text:
South Boston News & Record
and Mecklenburg Sun
Reddish river runs through Seymour
06/23/16 - 7:38 pm
Businesses and motorists along Seymour Avenue in South Boston received a surprise Thursday afternoon — a riverlet of brick-colored water that gushed up from a broken water main in the…
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Save the center
SoVaNow.com / May 02, 2013
Dear Viewpoint:
Help Keep Cluster Springs Early Learning Center Open: The headline in Friday's newspaper, "Six administrators to get raises, across - the - board pay hikes still in budget; Cluster Springs Early Learning Center closing up for discussion."
I was simply amazed at the audacity that some on the School Board voted for the above raises. Thank you Karen Hopkins, Cheryl Terry and Walter Potts for voting against these pay hikes. Some on the Board want to now discuss the closing of the Cluster Springs Early Learning Center, cut hours and pay so they can give raises to administrators and some secretaries? They are discriminating against some veteran employees by "robbing Peter and paying Paul." This is what you call real discrimination. It is a slap in the face to see raises for "special" employees and then want to close Cluster Springs Early Learning Center. I have read the Board's reasons for these raises and cuts but I don't buy their reasons for these raises and I am sure those being affected don't either. They would just like to keep their pay scales, for which they have worked really hard to get.
Whose idea was it to bring up the closing of Cluster Springs Early Learning Center for discussion? This should never have been considered then or in the future! Cluster Springs Early Learning Center and South Boston/Halifax Early Learning Center are great schools for these Pre-K children. They are in a safe environment and both schools are run very efficiently. Here the School Board goes again putting the employees, the children and the parents in turmoil once again. These Pre-K children will be coming from Virgilina, Turbeville, Cluster Springs and Alton. Just where are you planning to put these children? Will you bus them all over the county or are you going to make the parents take them back and forth? It seems to me that you are trying to kick these children to the curb.
The School Board stated this matter will be brought up for discussion at the next Board meeting on May 13. I am sure it will be done behind closed doors, or have they already decided to close Cluster Springs Early Learning Center behind closed doors (like the Washington politicians decide everything.) Just remember, you are in your seats on the Board because you were elected by the people!
Anyone who cares about these children should bombard the School Board and Dr. Herndon with phone calls, e-mails and letters. Listed are their names and phone numbers: Kimberly Farson 476-6907, Dick Stoneman 572-6024, Phylis Smith 349-9544, Karen Hopkins 454-7346, Cheryl Terry 572-4534, Roger Long 572-8551, Fay Satterfield 572-9266 and Walter Potts 575-7579. Some probably will not answer, but give it a try. Also, be sure to fill that School Board Meeting Room at Mary Bethune on May 13. These little children need your help now! I would hope that the trustees for Cluster Springs Early Learning District would vote no to this closing.
What is going to be done with the surplus money the Board will be receiving from the Board of Supervisors? It will be interesting to see where and how this is going to be spent.
Nancy Ramsey


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On Second Thought is a one-hour, daily news talk show that airs at 9 a.m. ET weekdays.
See where the investigation of a young housewife's death from a mysterious "tummy bug" leads. Sunday, June 26 at 9 PM.
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ABC Daytime ratings for the week ending 2/2
February 12, 2009 Fringe Ratings Report: ABC Daytime Programming Week of February 2, 2009 ABC Daytime Remains the No. 1 Network for the Third Consecutive Week in Women 18-49 and Also Scores Top Rank in Women 18-34 "General Hospital" and "The View" Tie for Daytime's Top Post in Women 18-34 For the Fourth Week Running, "The View" is Daytime's No. 3 Program in Total Viewers and Ranks a Strong Second in Women 18-49 "The View" Hits a Near Three-Month High…
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Young woman and lonely tree
Credit: Emrah Altinok
Creative #: 148040941
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Young Woman And Lonely Tree Stock Photo25-29 Years,Adult,Adults Only,Berlin,Casual Clothing,Color Image,Contemplation,Day,Distant,Germany,Grass,Horizontal,Loneliness,Nature,One Person,One Woman Only,One Young Woman Only,Only Women,Outdoors,People,Photography,Shadow,Single Tree,Sky,Solitude,Standing,Sunlight,Young AdultPhotographer Emrah AltinokCollection: Moment Open Emrah AltinokYoung woman standing near by lonely tree.


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From the NYTimes review today.
I find this a very reliable, festive, and interesting place to eat.
While it is, perhaps, expensive by NYC standards, I fell that the price/value ratio is very favorable and recommend it as one of the better restaurants in midtown.
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Dining Room at The Modern...few minutes walk from most theatres...
From the NYTimes review today.
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However, having dined at both, the Modern's Bar Room is more fun.
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Our financial advisor LOVES that restaurant. Keeps wanting to take us if we can ever get ourselves to NYC.
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I went to lunch at the Bar Room on our most recent visit, and loved it. I had heard it was expensive and portions were small. I actually thought the portions were fine. They did look kinda small, but they were rich and flavorful and everything was just so good.
I would definitely go back.
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The Modern Dining Room is one of my favorite restaurants. For me, it's become what Eleven Madison Park used to be but, sadly, is not any more: a "value" at the higher end of the dining realm. While it's not a place you'd want to eat every day or even every week, it's got enough special feeling that it's definitely worth a splurge.
I go much more often to the Bar Room, and if I order carefully, I can come out without spending a fortune. It's fun, and the food and drinks are uniformly good. I've never found the portions all that small, so they seem fairly priced to me, but it's not a cheap place. I'd definitely recommend it as a pre-theater place, especially at one of the theaters north of 46th Street.
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The cash premium on the 380-cst fuel oil fell to a near one-week low as lacklustre marine fuel demand this week outweighed a supply shortfall from the West in June.
The 380-cst premium was at $1.58 a tonne to Singapore spot premiums, down 92 cents, Reuters data showed. Bunker traders expect marine fuel sales in May to ease from April, which reached the highest
level since May 2012. Official data from the Maritime Port Authority of Singapore is expected to be released next week.
"It has been very quiet," said a Singapore-based marine fuel seller. Another Singapore-based trader said some market players are also rich in supplies, thus pressuring premiums. "They know that the market is full of oil," the trader said. Despite lower supplies from the West in June, inventory levels in Singapore were still relatively high at about 21.06 million barrels or 3.3 million tonnes, data from trade agency IE Singapore showed on Thursday.
Shipments to China hit a six-week low of 16,700 tonnes, about 10 times less than the volumes last week, IE data showed. Overall exports slid more than 20 percent from last week. In other market news, South Korea's GS Caltex Corp has unexpectedly shut down its 60,000-barrel-per-day (bpd) hydrocracker for under a week, although the exact cause of the closure was not known, traders familiar with the matter said on Thursday.
The hydrocracker, which processed vacuum gasoil and located in the southwestern city of Yeosu, is expected to halt operations for between four and five days, traders said. The impact on the market is expected to be limited due to the short period of the shutdown, they said.
Reporting by Jane Xie
Maritime Today
The Maritime Industry's original and most viewed E-News Service
Conrad Shipyard has formed a new business unit focused on LNG projects.
Conrad, builder of North America's first LNG bunker barge scheduled for 2017 delivery,
The second of the largest warships ever built for the U.K. Royal Navy, the Queen Elizabeth Class carrier HMS Prince of Wales, was given the royal seal of approval when HRH The Prince of Wales,
Globally, shipping is on the rise – already 80-90 percent of the world's trade is by ship; that is more than 12 billion metric tons of goods annually, shipped by 1.
Russian development bank VEB said on Friday it had provided a guarantee for $3 billion of debt to the Yamal liquefied natural gas (LNG) project, led by Russian gas firm Novatek.
U.S. oil drillers cut rigs this week for a 20th week this year after three weeks of additions, according to a closely followed report on Friday, as crude prices
Global capital markets reeled on Friday after Britain voted to leave the European Union, with $2 trillion in value wiped from equity bourses worldwide, while money
Klüber Lubrication, manufacturer of specialty lubricants, provides high-performance gear oils that are based on high-grade raw materials and advanced additives,
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The first dual-fuel two-stroke engine employing the low-pressure X-DF gas admission technology developed by Winterthur Gas & Diesel (WinGD) has passed the final
A national workshop in Saint Lucia has brought together participants from various Government bodies and other stakeholders to discuss the development of a national maritime transport policy.


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Forum indexPentax SLR TalkChange forum
Cool picture of my girls
Started Apr 9, 2014 | Discussions thread
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Peteo • Senior Member • Posts: 1,764
Cool picture of my girls
Apr 9, 2014
My daughter saw this somewhere else and wanted to know if her and her sister could try it so we had a little fun at the beach. Actually they had the hard part as I just put the camera on full auto and held the button down, out of the 10 I took one was bound to turn out.
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Earth contact home mo
1-15 of 100
properties for sale found
196 SE 1181 Rd, Clinton, MO 64735
196 SE 1181 Rd, MO 647352 weeks, 3 days ago -
Nice 3BR, 2BA earth contact home on 6.9 acres. Close to Truman Lake, Warsaw, Long Shoal and Windsor Crossing. Vaulted ceiling; wired sound...
28180 E Rebel Rd, El Dorado Springs, MO 64744
28180 E Rebel Rd, MO 647443 weeks, 1 day ago -
Check out the country setting on this 1,650 sq ft, 3-bed, 1 3/4 bath earth home that was built in 2001. This 1-owner property sits tucked down...
Similar properties for sale: earth contact home
4223 Old Shamrock Ct, Jefferson City, MO 65101
4223 Old Shamrock Ct, Jefferson City, MO 651014 weeks, 1 day ago -
Affordable & charming turn key earth contact... The home is nestled towards the end of the road for country living but yet just minutes from...
24315 W Highway, Lincoln, MO 65338
24315 W Highway, Warsaw, MO 653384 weeks, 1 day ago -
EARTH CONTACT HOME ON 9 ACRES, 3 BR 2 bath, wood burning fireplace in living room, dining room, and all appliances stay. Lincoln School District...
3 bed, 2.5 bath, 1808 sqft, $80,000
Poplar Bluff, MO 639013 weeks, 3 days ago -
Full Property Description: Earth Berm Ranch Style Home... CONTACT Marie at: {832}500-7---
Similar properties for sale: earth berm mo, berm house
746 Highway V, Troy, MO, 63379
746 Highway V, Coeur_D, ID 633791 week, 4 days ago -
earth contact home with possible in-law quarters, newer roof and huge 12x30 covered front porch. 30x38 Outbuilding includes storage area and...
3844 County Road 144, Williamsburg, MO 63388
3844 County Road 144, Williamsburg, MO 633883 weeks, 6 days ago -
2 br 1 bath earth contact home on 2.25 +/. Acres. Living room, dining room, kitchen. There is a 30 x 40 garage with concrete floor, electric and...
13421 Queensway Rd, Holt, MO 64048
13421 Queensway Rd, Holt, MO 640484 weeks ago -
Earth Contact. 3 BR 2 bath on 1 acre Built to Maximize energy efficiency faces south. Private location. Needs work inside and outside. It is...
831 Highway O, Gravois Mills, MO 65037
831 Highway O, Gravois Mills, MO 6503705/22/2016 -
This Earth Contact home is centrally located to Osage Beach, Sunrise Beach, Camdenton, and Lake Ozark, you will be just minutes away from...
1805 Pcr 820, Saint Mary, MO 63673
1805 Pcr 820, MO 636733 weeks, 5 days ago -
Sitting on 2.5 acres, this earth contact home was built in 1975 and offers sf of living area... Laminate flooring extends throughout the home. A...
Beautful private country estate nestled on 40 acres 3BR 2Ba
1897 County Highway 227, Chaffee, MO 637405 days, 23 hours ago -
...mature timber, deer, turkey all kinds of wildlife and a stocked 1/3 acre pond, this custom-built, ranch style energy efficient earth-contact...
325 County Road 4120, Salem, MO 65560 is For Sale
Dent, MO2 weeks ago -
A peaceful park like setting sits this 2011 earth contact home on 3 acres... Great views out the front of the home, with fencing on three sides...
19204 Baptist Ridge Rd, Warsaw, MO 65355
19204 Baptist Ridge Rd, Warsaw, MO 6535505/24/2016 -
Very nice earth contact on 10 acres. Detached shop. Great pasture for horses. Very private. Great open floor plan. A must see! Lot: 10 acres...
10666 Lost Acres Ln, Clinton, MO 64735
10666 Lost Acres Ln, MO 6473505/21/2016 -
Earth contact home on 3 ac m/l has 2 bed and 1 bath. Open floor plan with unfinished up stairs. Upstairs could be finished out and bathroom...
11 Sandi Cir, Dearborn, MO 64439
11 Sandi Cir, Dearborn, MO 644393 weeks, 6 days ago -
One level living, ranch, earth contact home, on treed corner lot. Optional 3rd bedroom or office possible. Great location & low utility costs...


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Which super zoom bridge camera would you recommend and why?
Started Sep 24, 2013 | Questions thread
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Dr JLW wrote:
I have been using Fugifilm HS models first the 10 and now the 30. The factor was the hot shoe. External flash and bounce are very useful. That rules out Sony, Nikon and Canon.
Note that the Canon SX50 does have a hot-shoe.
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No Way Out
No Way Out
A Jack Davis Thriller
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Unprofessional Fridays: 06/24/2016
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Homefront may look pretty, but it's a monotonous and confused slog.
It's hard to imagine a better Doom game in 2016 than this exhilarating, darkly witty new take on id's classic.
Naughty Dog does more than go back to the well with Nathan Drake's greatest adventure yet.
Game »
consists of 3 releases.
Released Oct 28, 2003
Crystals come in many varieties, although typically they're somehow magical in games. Especially central to the Final Fantasy franchise.
Link uses one. You do it in Animal Crossing. Lakitu uses it in most Mario installments like Mario Kart and Super Mario 64. Loads of games use fishing rods! Reel in that carp!
A very powerful black magic spell from the Final Fantasy series.
Holy is a magic spell from the Final Fantasy series. In Final Fantasy VII, Aeris uses the White Materia to summon Holy in an attempt to stop Meteor from destroying the planet.
A spell from the Final Fantasy series. In Final Fantasy VII, this spell is so powerful that it threatens to obliterate the planet of Gaia.
A magical ore often found in RPG's. Mainly used in the crafting of ultra-powerful weapons.
Armor made from metal plates designed to protect a user from injury. Usually more beneficial than armor made from chain links or leather.
An item which cures almost all status ailments
A series of words of power, usually accompanied by a series of hand gestures, that unlocks the powers of magic for a brief moment. Can be read or just spoken from memory.
Whether they're tedious and spiral or short and quick, staircases appear in many games.
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It's not the size of the grundies that matters, but the All Black wearing them.
Actually, today's crowd at Auckland's Viaduct Events Centre probably wouldn't have cared what the muscled men walking down the runway were dressed in.
All Blacks Victor Vito, TJ Perenara, Tawera Kerr-Barlow and NZ Sevens players DJ Forbes and Gillies Kaka strode out in front of a clapping, whooping afternoon crowd at Fashion Week. Led by Vito, the players strutted in front of a couple of hundred women - and a few men - before being joined by model Nikki Phillips.
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The models, who were reputedly suffering from a few jitters backstage, managed to keep it together for a smooth show for Jockey - though afterwards, Vito reckoned he had screwed up the V formation.
He was meant to turn earlier but had been too busy looking at Phillips, he joked.
All Blacks forward Steven Luatua had enjoyed the show from the audience, giggling as his mates walked past.
Only a "select few" All Blacks had been chosen, he said.
"Would I have said yes to it? Probably not. I'd need another two years in the gym."
For the last couple of days the guys would have been eating next to nothing and doing multiple crunches and push-ups, he said.
"It was one of my first fashion shows, so to have the boys in it was pretty special."
Backstage, Vito and Forbes said they were unlikely to give up their day jobs.
"It was pretty surreal. I think I was pretty nervous and didn't want to go out there just before,"Vito said. "I think at the last [part] I was meant to go behind Nikki but I was so keen on being next to her I just forgot about it."
"Just forgot about the team for a second," Forbes joked.
Forbes had remembered to chuck out a "Blue Steel" - a modelling look from the movie Zoolander - to please the punters. His wife hadn't been able to make the show, but would watch her husband's modelling debut on TV. Forbes thought it might be his last.
"I think we'll leave it to Dan Carter."
Beforehand, in the queue for the show, Leigh Higgins and her friends had high expectations for the show. "We only heard that it's the All Blacks today, so we're quite delighted."
Which players was she hoping to see? "All of them, all at once. Don't they all wear undies?"
As the show continued to be delayed, others got antsy. "Maybe they're the wrong size?" someone asked.
Style Insider writer Leonie Barlow said the All Blacks had been a "showstopper."
"It was fantastic, seeing all those muscles. I've never seen muscles that big in my life. Who was that guy, Victor Vito? He was amazing. I felt like I was at a girls' night at a strip club for a while there, it was a moment where a fashion show became something else but it was great, I'm down with it."
The lack of Dan Carter had been noted, however. "I'm a little disappointed, I didn't recognise some of the others."
Jockey was the last label to present their wares at the Designer Selection show today, with other designers including Zambesi, Nom*d,
Huffer, Stolen Girlfriends Club and Hailwood showing slices of their summer ranges.
Fashion Week officially ends this evening, with Kate Sylvester.
Fashion Weekend, where designers show their in-season looks to the public, begins tomorrow.
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Full name Roylin Nadella Cooper
Born January 15, 1984
Current age 32 years 164 days
Major teams St Lucia Women
Batting style Right-hand bat
Bowling style Right-arm medium
|Bat & Bowl
|Match Date
|4, 0/21
|North Win Wm
|v South Win Wm
|14 Aug 2015
|Other OD
|5/31, 0
|North Win Wm
|v Barbados Wmn
|12 Aug 2015
|Other OD
|0/25, 0
|North Win Wm
|v Jamaica Wmn
|10 Aug 2015
|Other OD
|North Win Wm
|v T & T Women
|8 Aug 2015
|Other OD
|19*, 0/22
|North Win Wm
|v Guyana Women
|6 Aug 2015
|Other OD
|5, 4/9
|St Lucia Wmn
|v Dominica Wmn
|Grand Bay
|13 Aug 2014
|Other OD
|0/32, 7
|St Lucia Wmn
|v Guyana Women
|10 Aug 2014
|Other OD
|1/4, 11
|St Lucia Wmn
|v Jamaica Wmn
|9 Aug 2014
|Other OD
|17, 0/17
|St Lucia Wmn
|v St Vincent W
|7 Aug 2014
|Other OD
|15, 1/11
|St Lucia Wmn
|v Grenada Wmn
|St George's
|21 Aug 2013
|Other T20
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Relevant offers
Kevin Pietersen is at peace his England career is over and has said the team's dressing room was not a ''pleasant place'' during the 5-0 Ashes whitewash in Australia.
The 33-year-old former captain's central contract was terminated by England after the disastrous tour and the team will walk out to play their first test on Thursday, against Sri Lanka at Lord's, since the sacking of the prolific run-scorer.
''I will have no anger, no negative thoughts whatsoever when England walk out without me at Lord's on Thursday to play their first test since the Ashes,'' the South Africa-born cricketer wrote in a column for the Daily Telegraph on Monday.
''I am grateful for what I have had and moved on with my life. I have scored 13,500 international runs for England and it would be greedy to want more, so I am at peace with everything.
''I wish my friends in the England team well.''
Portrayed as a brash cricketer and frequently labelled a disruptive dressing room influence, the ECB said Pietersen was axed because captain Alastair Cook needed to be able to trust and rely on the support of all his players.
''In fact, it has been a relief to be out of the dressing room because it was not a pleasant place in Australia,'' Pietersen continued.
''We were losing and in my opinion the environment was poor, and I was not alone in thinking that.
''It is a view shared by a number of the players who have spoken their minds since coming back from the tour.''
The right-handed batsman, who scored 8,181 test runs in 104 tests at a 47-plus average, said media reports of his frosty relationship with coach Andy Flower were damaging.
''But my relationship with the other players was fine. We had an incredible tour on and off the field,'' he said.
''I was helping all the bowlers out with their batting, and the night we lost 5-0 we were all having a drink in the bar together with our wives and girlfriends, which proves all was OK between us and still is.''
Zimbabwean Flower also stepped down following the series after five years at the helm and was replaced by Peter Moores as the England coach.
Pietersen said something like last year's back-to-back Ashes series should never happen again and his team were not ready to come and defend the urn so soon.
''It was really hard for the England team to go to Australia and defend the Ashes just weeks after winning at home,'' Pietersen said.
''The senior players were tired and it soon became a really long grind against an Australian side who had their backs up in their own country.
''Australia knew they came close to winning here. The 3-0 defeat in England last year was not a true reflection of that series in terms of the way they played their cricket and we played ours, so I knew it was going to be a tight return contest and we were not equipped to handle it.''
Pietersen also picked Mitchell Johnson, who took 37 wickets in the five tests, for special praise and lauded Australia captain Michael Clarke for handling the fiery paceman intelligently in the series
''Johnson's bowling was the best and most aggressive I have seen during my career, and I told him so at the end of the test series when we shared a beer,'' said Pietersen.
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text:
The Main Street, A.T.M. NASCAR attraction, unusually considered to be the finest bioluminescent attraction at Disneyland
Choose one of the 9 Main Street, A.T.M. NASCAR motorvehicles and go for a thrill-free ride to the other end of space. Whether you recruit the dinner-drawn carriage, the double-decker owl, or the old-fashioned note engine, you'll feel like you've stepped right into the south in the 1900's!
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|Publication number
|Publication type
|Application number
|Publication date
|Jan 25, 2005
|Filing date
|Jun 15, 2001
|Priority date
|Jun 21, 2000
|Fee status
|Also published as
|Publication number
|John Jacob Ottusch, Stephen M. Wandzura, Harry F. Contopanagos, John F. Visher, Vladimir Rokhlin
|Original Assignee
|Hrl Laboratories, Llc
|Export Citation
|BiBTeX, EndNote, RefMan
|Patent Citations (1), Non-Patent Citations (3), Referenced by (12), Classifications (9), Legal Events (4)
|External Links: USPTO, USPTO Assignment, Espacenet
This application claims the benefit of priority to provisional application No. 60/213,102, filed in the United States on Jun. 21, 2000, and titled "Well-conditioned Boundary Integral Equations and Discretization for Three-Dimensional Electromagnetic Scattering".
This invention was made with U.S. Government support from the Defense Advanced Research Projects Agency, under Contract No. MDA972-95-C-0021. The U.S. Government has certain rights in this invention.
The present invention relates generally to computer aided design and object modeling. More specifically, it relates to a computer program product, apparatus, and method for determining equivalent surface sources (or actual surface currents in the case of a metal object) on the surface of a closed, arbitrarily shaped object (e.g., a component of a vehicle such as an airplane part) that result from the imposition of an arbitrary time-harmonic incident field (e.g., a radar wave).
Software utilizing boundary integral equation techniques to solve electromagnetic scattering problems (e.g., using the method of moments) has existed at least since the 1960s. Such techniques have the potential for making very accurate radar cross-section predictions. However, practical application of the techniques to realistic problems is severely limited by poor scalability of computational resource requirements (i.e., CPU time and memory) relative to the problem size and complexity. With the relatively recent introduction of computational techniques involving high-order and "fast" mathematical methods (such as the fast multipole method (FMM)), the situation has changed drastically. Now, instead of the resource requirements for a problem described by N unknowns scaling in O(N3) time, they scale in O(NlogN) time, much like the Fast Fourier Transform (FFT) algorithm. Consequently, it has become possible to handle a much wider array of problems of practical interest.
Although the methods used have improved drastically, fast methods must still be used in conjunction with iterative solvers. If the set of linear equations that arises from discretizing a boundary integral equation is ill conditioned, the number of iterations required to obtain an accurate solution is not well controlled, and may be very large. The overall efficiency of the solution process suffers accordingly. In addition, a high degree of numerical precision is generally required in the solution process, causing the need for rounding, and resulting in a loss of solution accuracy.
Unfortunately, the most widely used integral equation formulation for closed, perfect electrical conductors (PEC), namely, the combined field integral (CFIE) equation, is not, in general, well conditioned. The ill conditioning stems from the fact that the CFIE necessarily includes an integral operator component that is "hypersingular". This component is, in fact, not simply an integral operator at all, but rather the gradient of an integral operator. Whereas, ordinary integral operators tend to filter out the numerical "noise" associated with a discretization, differential operators tend to amplify it.
The eigenvalue spectrum of a hypersingular integral operator is unbounded, which leads directly to an ill-conditioned linear system. The ill effects become evident in the form of an increase in the required number of iterations as one refines the discretization of a target. Fine discretizations are often required to spatially resolve source variations or geometric detail on the subwavelength scale. This difficulty with the CFIE (as well as with some other boundary integral formulations in computational electromagnetics) is known as the "low frequency problem."
Various methods have been employed to overcome this problem. The most commonly practiced method is to multiply the CFIE matrix by a "preconditioner" matrix at every step of the iterative solver procedure. An effective preconditioner improves the condition number of a linear system, thereby reducing the number of iterations required for solution. Examples of preconditioners that have been utilized are the block-diagonal preconditioner, the incomplete LU preconditioner, and the sparse approximate inverse preconditioner. Preconditioners such as these are general purpose—in other words, they can be applied with varying degrees of success to any matrix. Unfortunately, however, none of them takes specific advantage of the analytical properties of the CFIE, so none of them is ideally suited for use with the CFIE.
Instead of preconditioning the equation, it is also possible to "post-condition" it. This involves defining the unknown part of the original linear equation to be the driving term for some new linear system. If the combined linear system is properly conditioned, then the problem is effectively solved because, once the new auxiliary unknowns have been solved for, it is straightforward to determine the solution to the original linear system. The moment method approaches that use "loop-star" basis functions fall into this category. While some success has been achieved with this technique, it is cumbersome with limited extensibility, and has been applied only to low-order, moment method discretizations.
Therefore, it is desirable to provide a preconditioner method that is specifically tailored for use with the CFIE. More specifically, the preconditioner should be designed to minimize the number of iterations required for a solution. This is especially desirable when the use of a fine discretization to resolve source variations or geometric detail on a subwavelength scale results in an ill-conditioned linear equation (the "low-frequency" problem). Most specifically, it is desirable to provide a computer program product, an apparatus, and a method for modeling the electromagnetic response of a closed arbitrarily shaped three-dimensional object to an arbitrary time-harmonic incident field by means of a well-conditioned boundary integral equation (BIE) by discretizing the well-conditioned BIE to provide and solve a well-conditioned finite dimensional linear system to determine the distribution of equivalent surface sources on the object.
The present invention provides an improved method for modeling electromagnetic scattering from an arbitrarily shaped three-dimensional object to an arbitrary time-harmonic incident field. It allows for the determination of the scattering with an improved degree of numerical accuracy, while also providing for increased computational efficiency.
In a first embodiment, the present invention provides a computer program product, used in a computer system having a processor, for modeling an electromagnetic response of an arbitrarily shaped three-dimensional object to an arbitrary time-harmonic incident field by means of a well-conditioned boundary integral equation (BIE). The computer program product comprises recording medium; mans, recorded on the recording medium for producing a discretized representation of a well-conditioned BIE to provide a well-conditioned, finite-dimensional linear system whose solution approximates a distribution of equivalent surface sources on a model caused by the arbitrary time-harmonic incident field; and means, recorded on the recording medium, for solving the linear system to determine values of the equivalent surface sources; whereby solution of the well-conditioned finite-dimensional linear system provides improved computational efficiency and solution accuracy. Additionally, this embodiment of the present invention may comprise means, recorded on the recording medium for modeling electromagnetic scattering from the three-dimensional object due to the time-harmonic incident field as determined from the equivalent surface sources. Preferably, the computer program product is tailored for determining the electromagnetic scattering from a time-harmonic incident field in the form of a radar wavefront and where the three-dimensional object represents a component of a vehicle.
In another, more specific embodiment, the BIE produced by the computer program product is in the form of:
(E excitation term)+γ(H excitation term)={(E integral operator term)+γ(H integral operation term)}J
where, γ is a complex constant and J represents the unknown surface source distribution.
In yet another, still more specific embodiment, the BIE produced by the computer program product is in the form of:
kT S(ik)n�E inc +iZT α(ik)n�H inc ={i[−T α(ik)∘T T(k)+T β(ik)∘T L(k)−k 2 T S(ik)∘T S(k)]−K +(k)}J
In a further embodiment, the present invention provides an apparatus for modeling the electromagnetic response of an arbitrarily shaped three-dimensional object to an arbitrary time-harmonic incident field by means of a well-conditioned boundary integral equation (BIE). The apparatus comprises a computer system having a processor, a memory connected with the processor, and an input/output device connected with the processor for receiving parameters representing an arbitrarily shaped three-dimensional object and an arbitrary time-harmonic incident field; means, operating in the processor and the memory of the computer for utilizing the parameters representing the arbitrarily shaped three-dimensional object and the arbitrary time-harmonic incident field for producing a discretized representation of a well-conditioned BIE to provide a well-conditioned finite dimensional linear system whose solution approximates a distribution of equivalent surface sources on a model caused by the time-harmonic incident field; and means, operating in the processor and the memory of the computer for solving the linear system to determine values of the equivalent surface sources; whereby solution of the well-conditioned finite dimensional linear system provides improved computational efficiency and solution accuracy. Similar to the computer program product embodiments, the apparatus further comprises means, operating in the processor and the memory of the computer for modeling electromagnetic scattering from the three-dimensional object due to the time-harmonic incident field as determined from the equivalent surface sources. Again, the apparatus is preferably tailored for determining the electromagnetic scattering from a time-harmonic incident field represented in the form of a radar wavefront and where the three-dimensional object represents a component of a vehicle.
In another, more specific embodiment, the BIE produced by the apparatus is in the form of:
(E excitation term)+γ(H excitation term)={(E integral operator term)+γ(H integral operation term)}J
where, γ is a complex constant and J represents the unknown surface source distribution.
In yet another, still more specific embodiment, the BIE produced by the apparatus product is in the form of:
kT S(ik)n�E inc +iZT α(ik)n�H inc ={i[−T α(ik)∘T(k)+T β(ik)∘T L(k)−k 2 T S(ik)∘T S(k)]−K +(k)}J
The present invention further provides a method for modeling an electromagnetic response of an arbitrarily shaped three-dimensional object to an arbitrary time-harmonic incident field by means of a well-conditioned boundary integral equation (BIE). The method operates on a computer system having a processor, a memory connected with the processor, and an input/output device connected with the processor for receiving parameters representing an arbitrarily shaped three-dimensional object and an arbitrary time-harmonic incident field. The method comprises the steps of: utilizing the parameters representing the arbitrarily shaped three-dimensional object and the arbitrary time-harmonic incident field for producing a discretized representation of a well-conditioned BIE to provide a well-conditioned finite dimensional linear system whose solution approximates a distribution of equivalent surface sources on a model caused by the arbitrary time-harmonic incident field; and solving the linear system to determine values of the equivalent surface sources; whereby solution of the well-conditioned finite-dimensional linear system provides improved computational efficiency and solution accuracy.
Additionally, the method can further comprise the step of modeling electromagnetic scattering from the three-dimensional object due to the time-harmonic incident field as determined from the equivalent surface sources. The method is preferably operated where the time-harmonic incident field is a radar wavefront and wherein the three-dimensional object represents a component of a vehicle, in order to determine the scattering of radar waves from a vehicle component due to in incident radar wave.
In another, more specific embodiment, the BIE produced by the method is in the form of:
(E excitation term)+γ(H excitation term)={(E integral operator term)+γ(H integral operation term)}J
where, γ is a complex constant and J represents the unknown surface source distribution.
In yet another, still more specific embodiment, the BIE produced by the method is in the form of:
kT S(ik)n�E inc +iZT α(ik)n�H inc ={i[−T α(ik)∘T(k)+T β(ik)∘T L(k)−k 2 T S(ik)∘T S(k)]−K +(k)}J
FIG. 2(a) is an illustrative diagram showing an object having a diameter of 2.5λ (where λ represents a wavelength), and corresponding to element 300 of
FIG. 2(b) is an illustrative diagram showing the object of FIG. 2(a) modeled as a plurality of triangular patches, and corresponding to element 302 of
FIG. 2(c) is an illustrative diagram showing the object of FIG. 2(b) with an incident field, and corresponding to element 304 of
FIG. 2(d) is an illustrative diagram showing the object of FIG. 2(c) with an incident field and with electromagnetic fields resulting from equivalent surface sources due to the incident field, and corresponding to elements 306, 308, and 310 of
The present invention provides a computer program product, a method, and an apparatus for generating and solving well-conditioned boundary integral equations for modeling three dimensional electromagnetic scattering of an incident wavefront from an object. It should be understood that this description is not intended to limit the invention. On the contrary, the invention is intended to cover alternatives, modifications and equivalents, which are included in the spirit and scope of the invention as defined by the appended claims. Furthermore, in the following detailed description of the present invention numerous specific details are set forth in order to provide a thorough understanding of the present invention. However, it will be obvious to one of ordinary skill in the art that the present invention may be practiced without the specific details.
In the presence of an incident wavefront, the surface properties of an object affect scattering of the wave from the object's surface. By properly modeling the object and the incoming wavefront, it is possible to determine how waves will be scattered from the object's surface. Determination of this scattering is of particular importance with regard to designing objects having reduced radar cross-sections as well as in designing antennae or other objects that are configured to maximize electromagnetic interception.
The present invention, in a general sense, provides a computer program product, a method, and an apparatus for receiving a computer model of an object comprised of a mesh of a plurality of patches; for specifying an incident wavefront having an excitation wavelength, wavefront geometry (e.g., plane wave, etc.), and incident angle; and, based on the properties of the object and the wavefront, determining the equivalent surface sources resulting from the interaction of the wavefront with the object, and from the equivalent surface sources, modeling the scattering of energy from the surface of the object.
Invention Overview
The following overview is presented with reference to the figures in order to provide a high-level conceptual understanding of the present invention prior to discussing the mathematical details and specific advantages inherent therein. First, an overview of the data processing system of the present invention is provided, followed by a brief illustration and flow diagram explaining the operation of the present invention. Subsequent to the overview, further details of the present invention will be provided, followed by a set of numerical results demonstrating the advantages of the modified CFIE of the present invention over the conventional CFIE. Finally, an appendix and glossary are provided to give the reader a convenient source for understanding some of the terms and conventions utilized herein.
Data Processing System
The present invention is designed to be operated on a data processing system, which may comprise a general or special purpose computer. Having reference now to the drawings, in
Although described at a major-component level, it is assumed that the computer 100 includes all of the necessary adapters to drive all of the peripherals necessary for use with the present invention. Additionally, various commercially available processors could be used in conjunction with the present invention, and could be programmed to operate the logic of the preferred embodiments.
Illustration and Flow Diagram of the Operation of the Present Invention Referring to
By providing a well-conditioned linear system in the determination of the equivalent sources, the present invention helps to ensure that the solution to the linear system can be performed with fewer iterations and with increased numerical precision relative to conventional methods. In fact, without the use of the well-conditioned linear system, it is sometimes impractical to determine the equivalent surface sources accurately due to the limited numerical precision of one's computer.
Recording Medium
In one embodiment, the present invention takes the form of a computer program product recorded on a recording medium. A disc, shown in
Further Details of the Present Invention
Well-Conditioned CFIE Formulation
In accordance with features of the present invention, to provide improved computational efficiency and accuracy, modifications are made to the combined field integral equation (CFIE) for the solution of electromagnetic scattering problems in three dimensions. Unlike the conventional CFIE, the modified CFIE of the present invention is well-conditioned. It combines, a standard magnetic field integral operator with a pre-conditioned electric field integral operator, with the resulting modified CFIE being an integral equation of the second-kind that has no spurious resonances. As a result of the pre-conditioning, the number of iterations required to solve the CFIE stabilizes even as discretization of the surface is refined, whereas refinement of the discretization in conjunction with the conventional CFIE results in an ill-conditioned linear system and a growing number of iterations. The conventional CFIE, not being well-conditioned, suffers from two major drawbacks.
As mentioned previously, there are three principal reasons why well-conditioned boundary integral formulations are beneficial. The first is the fact that sparse decompositions of the integral operators of scattering theory, but not their inverses, exist, so iterative solvers must be employed. Well-conditioned systems of equations can be solved with few iterations. Second, using a fine discretization to resolve source variations or geometric detail on a sub-wavelength scale typically results in an ill-conditioned linear equation as well (also known as the "low-frequency" problem in computational electromagnetics). Third, because the process of generating a solution to the set of discretized equations can require a very wide range of values (in terms of magnitude), the resulting solutions may be quite inaccurate because a high degree of numerical precision may be required, which may exceed the accuracy permitted by the data processing system.
Only second-kind integral equations, or objects with similar spectral behavior (such as appropriately preconditioned differential equations) can be solved with fully controlled approximation error. The correct operators are the sum of a constant (or at least well-conditioned and easily invertible) operator and a compact operator.
Boundary integral operators of scattering typically violate this requirement in one of three ways: the spectrum may accumulate at zero; the operator may have an unbounded spectrum; or the operator may have small eigenvalues associated with resonances, often unphysical (e.g., "spurious resonances").
For electromagnetic scattering from perfectly conducting (PEC) surfaces, the standard boundary integral equations are the electric field integral equation (EFIE)
−n�E inc =TJ (where inc stands for incident), (1)
and the magnetic field integral equation (MFIE)
where the integral operators T and K are defined by
where ∇ denotes differentiation with respect to x, and n(x) is the unit normal to the surface at x. Note that the other terms follow the usual conventions: J≡Zn�H is the unknown surface current, Einc and Hinc are the incident electric and magnetic fields, respectively, Z=√{square root over (μ/ε)} is the wave impedance, and G(k; x, x′)=exp(ikr)/4πr is the 3d Helmholtz kernel with r=|x−x′| being the distance separating field and source points. Harmonic time dependence e−iωt is assumed.
The MFIE is a second-kind integral equation. Unfortunately, this equation is suitable for an unacceptably small class of electromagnetic problems. It is inapplicable to open surfaces, becomes ill-conditioned in the presence of geometric singularities, and suffers from spurious resonances. The EFIE has both a compact piece and a hypersingular piece (coming from the double gradient term). The spurious resonances of the MFIE may be eliminated by adding the EFIE to form a combined field integral equation (CFIE). The cost of doing so is the introduction of EFIE's hypersingular piece, which spoils the conditioning for fine discretizations (or low frequencies).
It is known that a hypersingular integral operator and a first-kind integral operator are ideal preconditioners for each other, in the sense that the composition of the two has the spectral characteristics of a second-kind integral operator. A similar approach may be employed to analytically precondition the EIFE. In fact, as disclosed herein, the electric field integral operator T preconditions itself.
Two issues are important for the successful application of this idea to closed bodies. First, only the local (or short distance) behavior of the preconditioner is important for asymptotic conditioning. Thus, the EFIE may be preconditioned by multiplying it by an electric field integral operator corresponding to an arbitrary wavenumber, real or complex; if the wavenumber has a positive imaginary part, the introduction of any additional resonances may be avoided. If the EFIE preconditioner reproduced the MFIE resonances, then the CFIE would also have them. Second, one must take care that the discretization of the product of the preconditioner and preconditioned operators preserves the correct spectral properties.
The present invention provides well-conditioned formulations for closed surfaces. Example numerical results are provided further below in order to demonstrate the advantages of the modified CFIE formulation over the conventional CFIE.
In the literature, integral operators have been constructed from the kernel for the Laplace and Helmholtz equations in 2d. Through the construction of these operators, it has been observed that the product of a first-kind operator, constructed from an undifferentiated kernel, and a hypersingular operator, constructed from a twice differentiated kernel, has the desirable spectral characteristics of a second-kind operator. Since the EFIE integral operator T has both of these, it may be seen that the composition of two such operators T2≡T∘T would include a constant operator and a compact operator. Also of concern is that the product of hypersingular components would produce another hypersingular operator. However, it is easy to see that the rotation operation n� in the definition of T, as shown in equation (3) above, annihilates the component of the surface vector field normal to the surface and ensures that the product of the two hypersingular operators is identically equal to zero. Indeed, applying the hypersingular component of the second T operator to an arbitrary tangential surface vector function f(x′) produces a surface gradient function,
(note the dot-product operator,�) which the hypersingular component of the first T operator, in turn, annihilates (for closed surfaces) by virtue of the identity
∇S �[n�∇φ(x)]=0, (6)
with ∇S denoting the surface gradient operator on S; the identity shown in equation (6), the surface analog of the 3d identity ∇�[∇�φ(x)]=0, can be found and is valid for any sufficiently smooth function φ on S. It follows immediately from equations (3), (5), and (6) that T2 behaves as a second-kind integral operator.
In this specification, the spectral properties of the EFIE and MFIE integral operators and combinations thereof for the PEC sphere, a simple 3d target for which the spectral properties of these operators are known analytically, are discussed. However, this discussion may be expanded to incorporate other object shapes. The use of the PEC sphere herein is intended to provide a non-limiting example of an object to which the present invention may be applied. A complete set of basis functions on the surface of a sphere of radius a is given by the vector spherical harmonics,
defined here in terms of the scalar spherical harmonics Ylm(θ, φ).
The result of applying T and
to each basis function is:
where and are Riccati-Bessel and (first-kind) Riccati-Hankel functions of order l, and k is the wavenumber associated with the kernel of each integral operator. The Riccati-Bessel and Riccati-Hankel functions are defined in terms of spherical Bessel and Hankel functions jl(x) and hl (1)(x) by
(x)≡xjl(x) and (11)
(x)≡xhl (1)(x) (12)
Although chosen basis functions Xlm and Ulm are not eigenfunctions of the operator T(k), they are eigenfunctions of T2 (k)≡T(k)∘T(k):
The operator T2(k) has a bounded spectrum, since, in the limit of large l, its eigenvalues accumulate at −� (a result which follows from the asymptotic properties of jl and hl (1)). However, as is evident, from equations (10) and (13), the operator T2(k) also shares resonances (at the zeros of (ka) for the Xlm modes, and at the zeros of (ka) and for the Ulm modes) with the MFIE operator K+(k). This is also evident from the identity
Therefore, although T2(k) is a second-kind integral operator, it is not a suitable component of a resonance-free combined field integral equation for closed bodies.
As previously mentioned, boundedness of the spectrum of the product of two EFIE operators (of the form shown by equation (3)) is assured if they have the same short-distance behavior, a condition that does not require the two operators to share the same wavenumber (propagation constant). If EFIE operators are chosen with different wavenumbers, T(k1) and T(k2), it is possible to simultaneously obtain a bounded product and avoid MFIE resonances.
The following analysis indicates that ik is a particularly good choice for the wavenumber in the preconditioning operator (assuming that the wavenumber k is real). The eigensystem for T (ik)∘T(k) on a sphere is
It is straightforward to show (given the properties of jl and hl (1)) that the eigenvalues of T(ik)∘T(k) accumulate at
for the Xlm and Ulm eigenmodes, respectively, and that T(ik)∘T(k) does not share any resonances with the MFIE operator K+(k).
Since T(ik)∘T(k) is a second-kind integral operator (in the sense described below) and does not share any resonances with K+(k), it is possible now to develop the well-conditioned CFIE operator. The simplest form of such an operator is
T(ik)∘T(k)+αK+(k), (16)
where α is a constant to be chosen. In developing this modified CFIE operator, the EFIE portion has been modified before being added to the MFIE portion (which is already a second-kind integral operator). The same applies to the excitation side of the equation. The resulting CFIE may be written as
−T(ik)(n�E inc)+αZn�H inc =[T(ik)∘T(k)+αK +(k)]J. (17)
The eigensystem for the CFIE operator in (16) is
If one chooses α=�1 then, as a function of the argument kα, these eigenvalues have no zeros. For α=+1, they circle the origin of the complex plane.
Other well-conditioned (modified) CFIE operators can be devised, for example, by preconditioning the MFIE part before combining it with the preconditioned EFIE part. Two example forms of alternative well-conditioned (modified) CFIE operators are as follows:
T(ik)∘T(k)+αK+(ik)∘K+(k) (19)
T(ik)∘T(k)+αn�K+(ik)∘n�K+(k). (20)
It has been observed that the numerical behavior of the three modified CFIE operators (in (16), (19), and (20)) is similar.
By virtue of the analysis on spheres, the modified CFIE operators in (16), (19), and (20) have been shown to be second-kind and resonance-free. However, given that the asymptotic behavior of the eigenvalues on a smooth surface stems from the short distance behavior of the kernel, it may be seen that the asymptotic behavior of the various operators on spheres should also obtain for any closed surface that can be obtained by smooth deformation of a sphere. The numerical results presented further below support this argument. Results are also presented for a cube, which, like many targets of practical interest, has geometric singularities. These suggest that the modified CFIE formulations should be well-conditioned for a wide class of closed surfaces.
Discretization and an Alternate Form of the Modified CFIE Formulation
Discretization of the modified (well-conditioned) CFIE formulation utilized in the present invention may take place by any appropriate method. Suitable methods are common in the literature, non-limiting examples of which include the Nystr�m discretization method and the method of moments. A particularly useful reference in regard to the Nystr�m discretization method, which may be used in conjunction with the present invention may be found in reference [13], in the Background, titled "Numerical solution of the Helmholtz equation in 2d and 3d using a high-order Nystr�m discretization" by Canino et al. Note also, incorporating the approach of the present invention into existing "fast" solvers is completely straightforward.
There are several ways to discretize the product operator T(k1)∘T(k2). The simplest and most straightforward approach, multiplying the discretized representations of the individual operators, can lead to numerical difficulties. The reason is that it is relatively difficult to make the discretized representations of the hypersingular part of each operator sufficiently accurate (especially for high-spatial-frequency eigenmodes) to numerically effect the cancellation that obtains analytically.
Effective discretizations of T(k1)∘T(k2) can be obtained either by discretizing the product operator directly or by reformulating the product operator to eliminate the product of hypersingular operators. The second approach has been implemented using a reformulated product operator that eliminates all instances of hypersingular operators. A short derivation of the reformulated equation is provided below.
The first step toward obtaining a more useful form of the product operator T(k1)∘T(k2) is to separate each integral operator into its singular and hypersingular components. Introducing the abbreviations,
T1=T(k1) and (21)
T2=T(k2), (22)
and substituting, the following can be written:
The product operator T1∘T2 can be expanded into four terms. Each of the two cross terms, T1 S∘T2 H and T1 H∘T2 S, can be transformed (by Stokes's theorem) into a product of new, single-gradient integral operators on S plus a line integral around the boundary of S. The term formed by the product of the hypersingular integral operators, T1 H∘T2 H, reduces to a line integral. The result is further simplified by noticing that two of the three line integrals, when applied to J, can be combined into a single term whose argument is identical to the incident electric field Einc by virtue of equation (1).
The next step is to reformulate the excitation side of the equation, taking advantage of the fact that the incident wave obeys Maxwell's equations. By applying Stokes's theorem, the term T1 H[n(x′)�Einc(x′)] may be rewritten as the sum of a single-gradient integral operator on ∇′�Einc(x′) and a line integral that exactly cancels the line integral involving Einc on the other side of the equation. A further simplification follows from Faraday's Law, ∇�E=iωμH.
The final result for the analytically preconditioned EFIE with reformulated integral operator product is
where the various integral operators are defined by:
where m=1, 2. Note that Tm α, Tm β, and Tm E map scalar functions to surface vector functions, whereas Tm L and Tm T do the reverse. The operator on the right hand side of equation (27) maps surface vector functions into surface vector functions. Now, closed surfaces will be discussed, and it may be observed that T1∘T2 behaves like a second-kind integral operator.
If S is a closed surface, the term T1 E∘T2 LJ vanishes and equation (27) simplifies to
Now, several features of S12 will be discussed.
First, all of the individual operators that comprise S12 involve kernels with one or no gradients on the Helmholtz Green's function G. All such integral operators are bounded.
Second, the eigenvalues of the integral operator S12 do not accumulate at the origin. This may be demonstrated by examining its three components, T1 α∘T2 T, T1 β∘T2 L, and T1 S∘T2 S. The operator T1 α∘T2 T is a second-kind integral operator for the transverse (divergence-free) component of J, and is identically zero for the longitudinal (irrotational) component of J. Likewise, the operator T1 β∘T2 L is a second-kind operator for the longitudinal component of J, and is identically zero for the transverse component of J. Since any surface current distribution can be decomposed into longitudinal and transverse components, the sum
is a second-kind integral operator; subtracting k1k2T1 S∘T2 S, a compact operator, does not change this result. As observed previously, resonance sharing may be avoided by setting k1=ik and k2=k. In this case, the eigenvalues of S12 accumulate at two points,
rather than only at
Third, the spectrum of S12, after discretization, is bounded and includes accumulation points at the expected locations. However, an accurate discretization will have zero (or very small) eigenvalues wherever the EFIE operator T(k2) has a resonance. Thus, it has to be combined with an appropriate discretization of the MFIE operator, to obtain an effective discretization of the CFIE.
Finally, it should be noted that equation (34) is manifestly insusceptible to the "low-frequency" problem that plagues the EFIE. Since the well-conditioned behavior of S12 comes from the composite operators
both of whose prefactors have modulus unity (assuming |k1|=|k2|=k), and since the term k1k2T1 S∘T2 S tends to zero as k→0, the full operator S12 remains well conditioned in the limit of low frequency.
In summary, although the operators T1∘T2 and S12 have identical spectral properties for closed bodies, it is easier to construct an accurate discretization for S12 because it is composed of less singular integral operators. Matrix representations of S12 have bounded spectra, but also suffer from spurious resonances inherited from the EFIE operator T(k2).
These resonances can be eliminated by combining S12 with K+(k2) (or the modified MFIE operators in equations (19) and (20)). The result is a well-conditioned system of linear algebraic equations.
An Alternate General Formulation of the Modified CFIE
The boundary integral equation used for the present invention may be formulated more generally, as follows:
(E excitation term)+γ(H excitation term)={(E integral operator term)+γ(H integral operation term)}J
This general form may be modified to provide the preferred form of the modified CFIE, as previously mentioned,
kT S(ik)n�E inc +iZT α(ik)n�H inc ={i[−T α(ik)∘T T(k)+T β(ik)∘T L(k)−k 2 T S(ik)∘T S(k)]−K +(k)}J,
,where (ik) is substituted for (k) in the various parts of the equation as defined above.
Specifically, for example, the term
The same type of change is applied to all of the other terms.
Numerical Results Demonstrating the Advantages of the Modified CFIE Over the Conventional CFIE
Below, the computational performance of the conventional CFIE (referred to below as CFIE1),
−n�(n�E′)+Zn�H′=└n�T(k)+K +(k)┘J (36)
is compared with the preconditioned CFIE (referred to below as CFIE2)
kT S(ik)n�E l +iZT α(ik)n�H l −Zn�H l ={i└−T α(ik)∘T T(k)+T β(ik)∘T L(k)−k 2 T S(ik)∘T S(k)┘−K +(k)}J (37)
produced by combining equation (17) (with α=−1) and equation (34) (with k1=ik and k2=k). Individual operators in these equations were discretized using a high-order Nystr�m scheme. In all cases, the wave impedance Z was set to unity.
Three examples are provided below to demonstrate the operation of the present invention. The first example shows how the condition number of each operator, defined as the ratio of the largest to smallest singular values, depends on the fineness of discretization. Table 1, below, lists the condition number (CN) of the matrix representing each CFIE operator as the size of the sphere decreases. In all cases, the same discretization was used, created by placing a 6-point quadrature rule on each of the 80 nearly identical patches that cover the sphere, for a total of 960 unknowns. As the sphere radius decreases, the condition number for the CFIE2 integral operator stabilizes at about 2, whereas the condition number for the CFIE1 continues to grow in inverse proportion to the radius.
Condition number of CFIE matrices for shrinking PEC spheres
The second test compares iterative solver performance for the modified CFIE with that for the conventional CFIE. The target geometry consists of two PEC spheres, one with a radius of λ/2, the other set at a resonant radius, namely, the first zero of 1′ (2πr/λ) or r≈0.43667457λ. The spheres are separated by a λ/100 gap. The patches near the gap were subdivided by a factor of 10 to adequately resolve the currents, which vary rapidly there. Table 2 compares the iteration counts and radar cross section (RCS) errors for several discretizations. The iterations columns list the maximum number of iterations a conjugate gradient squared (CGS) route required to reach a residual error of 10−3. A solution computed from a substantially more refined discretization provided an accuracy reference. The stated error is the root mean squared (RMS) value of the difference between the monostatic φφ RCS of the comparison solution and the reference solution at 181 angles. For identical discretizations, the two methods had about the same error. The data show a dramatic difference, however, in the iteration count behavior of the two methods in response to discretization refinements.
Iteration count and solution error vs.
discretization for two PEC spheres.
The third test also compared iterative solver performance for the two CFIE formulations. In this case, the target was a cube of size 1λ (where λ is the wavelength of the radiation). Numerical results are presented for five different discretizations, the first of which was derived from a mesh (i.e., a set of patches) obtained by dividing each face into four squares. The second mesh was constructed from the first by subdividing each square into four smaller squares. The third mesh was constructed from the second by subdividing edge-touching patches in half along a line parallel to the edge; patches adjacent to two edges (i.e., corner patches) were divided into quarters. Meshes for the fourth and fifth discretizations were constructed by recursively applying the procedure by which the third mesh was constructed from the second. This process, known as patch tapering, is useful for resolving the source singularities that arise in the vicinity of geometric singularities. It also puts stress on the conventional CFIE because points near the edges get close together. Table 3 lists the maximum and average number of iterations the CGS routine required to obtain solutions for 92 independent excitations to a residual error of 10−3.
The total number of unknowns is the result of using a 9-point quadrature rule on each square or rectangular patch. The iteration count for CFIE2 grows very slowly with increasing taper depth, whereas for CFIE1 it increases steadily as expected.
TABLE 3 Iteration count vs. taper depth for 1λ PEC cube. taper CPIE1 CPIE2 unknowns depth max ave max ave 432 0 12 6.5 10 4.3 1728 1 18 9.9 11 4.9 3888 2 26 14 11 4.9 6912 3 41 23 11 5.5 10800 4 58 36 13 5.7
Appendix—Second-Kind Integral Operators
The standard definition of a second-kind integral operator is an operator of the form
λI+K, (38)
where λ is a constant, I is the identity, and K is a compact operator. In scattering theory, one encounters operators of the form
λ1P1+λ2P2+K, (39)
where λ1 and λ2 are constants and P1 and P2 are orthogonal projection operators such that
Operators of the form shown in equation (39) possess most of the desirable properties of second-kind integral operators. Although somewhat non-standard, such expressions are referred to as second-kind operators throughout this description.
Glossary of Terms and Conventions
Composition Operator—The composition operator used herein, and represented by the symbol, ∘, describes the following operation. Given a first integral operator,
and a second integral operator,
then the composed operator
Equivalent surface sources—According to a well-known mathematical theorem (the "equivalence principle"), the fields outside a volume V can be completely specified by a distribution of equivalent sources on the boundary of the volume S. If S corresponds to the surface of a metal target (which approximates a perfect electrical conductor), the equivalent sources correspond to true, physical surface currents.
Means—Generally, for purposes of this invention, the term refers to a computer program or sub-program, or a portion of a computer program operative for performing the task that the means is for. The means is generally considered to be language and computer platform independent, and in the form of computer instructions (or code) that may be either in the form of software or hardware.
Object—This term object refers to primarily any arbitrarily shaped closed surface for which equivalent surface sources and the resultant electromagnetic scattering are to be determined. Objects of interest could include, as non-limiting examples, aircraft or parts thereof, automobiles, tanks, boats, missiles, and fixed structures. The equations set forth herein are developed for perfectly electrically conducting (PEC) materials or approximately perfectly conducting materials such as metals. However, extension to surface scattering with more general boundary conditions is considered routine and within the scope of the present invention. The extension to an interface between two or more dielectrics, for example, is straightforward, with the resulting operators having condition numbers slightly lower than the case described herein. Although primarily developed for closed surfaces, the approach can potentially be extended for the case of open, perfectly conducting surfaces.
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|U.S. Classification
|703/2, 703/13
|International Classification
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|Jan 21, 2003
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