Near Fallujah, Iraq ? U.S. troops powered their way into the center of the insurgent stronghold of Fallujah on Tuesday, overwhelming small bands of guerrillas with massive force, searching homes along the city’s deserted, narrow passageways and using loudspeakers to try to goad militants onto the streets.

As of Tuesday night, the fighting had killed 10 U.S. troops and two members of the Iraqi security force, the U.S. military announced. The toll already equaled the 10 American military deaths when Marines besieged the city for three weeks in April.

U.S. officials issued no estimate of insurgent casualties, but one American commander said his battalion alone had killed or wounded up to 90 guerrillas.

As the offensive moved into a second full day, up to eight attack aircraft — including jets and helicopter gunships — blasted guerrilla strongholds and raked the streets with rocket, cannon and machine-gun fire ahead of U.S. and Iraqi infantry who were advancing only one or two blocks behind the curtain of fire.

Small groups of guerrillas, armed with rifles, rocket-propelled grenades, mortars and machine guns, engaged U.S. troops, then fell back. U.S. troops inspected houses along Fallujah’s streets and ran across adjoining alleyways, mindful of snipers.

A psychological operations unit broadcast announcements in Arabic meant to draw out gunmen. An Iraqi translator from the group said through a loudspeaker: “Brave terrorists, I am waiting here for the brave terrorists. Come and kill us. Plant small bombs on roadsides. Attention, attention, terrorists of Fallujah.”

Faced with overwhelming force, resistance in Fallujah did not appear as fierce as expected, though the top U.S. commander in Iraq said he still expected “several more days of tough urban fighting” as insurgents fell back toward the southern end of the city, perhaps for a last stand.

Some U.S. military officers estimated they controlled about a third of the city. Commanders said they had not fully secured the northern half of Fallujah but were well on their way as American and Iraqi troops searched for insurgents.

U.S. and Iraqi troops captured two key landmarks Tuesday — a mosque and neighboring convention center that insurgents used for launching attacks, according to a Los Angeles Times reporter embedded with U.S. forces.

“I’m surprised how quickly (resistance) broke and how quickly they ran away, a force of foreign fighters who were supposed to fight to the death,” Lt. Col. Pete Newell, a battalion commander in the 1st Infantry Division, told CNN.

Newell was quoted on CNN’s Web site as saying his battalion had killed or wounded 85 to 90 insurgents.

The move against Fallujah prompted influential Sunni Muslim clerics to call for a boycott of national elections set for January. A widespread boycott among Sunnis could wreck the legitimacy of the elections, seen as vital in Iraq’s move to democracy. U.S. commanders have said the Fallujah invasion is the centerpiece of an attempt to secure insurgent-held areas so voting can be held.

Prime Minister Ayad Allawi declared a nighttime curfew in Baghdad and its surroundings — the first in the capital for a year — to prevent insurgents from opening up a “second front” to try to draw American forces away from Fallujah. Clashes erupted in the northern city of Mosul and near the Sunni bastion of Ramadi, explosions were reported in at least two cities and masked militants brandished weapons and warned merchants to close their shops.

In Fallujah, U.S. troops were advancing more rapidly than in April, when insurgents fought a force of fewer than 2,000 Marines to a standstill in a three-week siege. It ended with the Americans handing over the city to a local force, which lost control to Islamic militants.

This time, the U.S. military has sent up to 15,000 U.S. and Iraqi troops into the battle, backed by tanks, artillery and attack aircraft.