That pattern seems strange for Dog Urine.  Have you checked for Cutworm or White Grubs.  Fence in a 6'x6' or similar area and cover with ground cover cloth then 1" crushed stone.  Train (make) the dogs do their business there.  Get a few rolls of sod.  Soak each in a 5 gal pail of water for a few minutes and cut out patches.  Dig out dead grass, place on new sod, cut matching patch and plant.